Why does compound interest exist? Background
My understanding is that compound interest arises in the following way:
The bank offers its clients some interest rate $r$ on an account with principal $P$ that yields $rP$ after some time $t_0$. But clients, not wanting to wait for $t_0$ to pass before seeing any returns, ask if they can instead have some fraction of the full return every time interval $t_0 / n$, where $n$ is some natural number. Reasonably, the bank agrees to pay out $r/n$ times the current balance every $t_0 / n$. The amount of money $A(t)$ in an account with principal $P$ after time $t$ is then the usual compound interest formula
\begin{equation}
A(t) = P\left(1 + \frac{r}{n}\right)^{\frac{nt}{t_0}} \label{comp-int}\tag{1}
\end{equation}
The Problem?
While the deal that the bank offers may seem reasonable for the bank, it results in the interest earned after time $t_0$ exceeding the original offer. Orignally, after $t_0$, the client would have a total balance of $(1 + r)P$. With compound interest, after $t_0$, the client would have a total balance of $A(t_0) = P(1 + r/n)^n$. The bank overpays (compared to its original promise) by a ratio of
$$
\frac{\left(1 + \frac{r}{n}\right)^n}{1 + r}
$$
which is bounded by
$$
1 \leq \frac{\left(1 + \frac{r}{n}\right)^n}{1 + r} \leq \frac{e^r}{1 + r} \label{bound}\tag{2}
$$
The problem, it seems to me, is that the offer is all downside for the bank. They give their clients money more often and end up giving them more money than the original offer. Why would a bank ever offer this option?
An Alternative?
If there were no other reasonable way to compute a partial return on an investment, then of course client demand would eventually force banks to make the compound interest offer.
But there is a readily available and (to me) perfectly reasonable alternative. Why not just compute the current balance as
$$
A(t) = P(1 + r)^\frac{t}{t_0} \label{alt}\tag{3}
$$
This has a few advantages. It doesn't require a new parameter $n$. It can be calculated continuously, but unlike the typical limit as $n$ goes to infinity (i.e. $Pe^{rt/t_0}$) it doesn't result in the bank overpaying. That is, $A(t_0) = P(1 + r)$, which was the original offer.
Speculation
I would speculate that the reason interest isn't calculated as in \ref{alt} is because it doesn't make sense to the mathematical layman. To the layman, the fair amount to be paid after time $t_0/n$ is $r/n$ times the current balance so that the new balance is $1 + r/n$ times the old balance — you just divide both by $n$. But the mathematically correct (I claim) way to evenly divide the payments would be to multiply the current balance by $(1 + r)^\frac{1}{n}$ every $t_0/n$, which seems more complicated and results in an effective interest rate less than $r/n$ per $t_0/n$. I can imagine that would be a tough sell to someone who doesn't understand the math. Instead the bank opts for the more approachable $1 + r/n$ factor, with the understanding that the deviation from the original offer is bounded (as \ref{bound} shows) and so an acceptable loss if clients are more likely to accept the offer.
Aside: musicians (or at least piano tuners) have to understand this math. The correct ratio between half-steps for equal temperament is $2^{1/12}$, not $1 + \frac{1}{12}$. While the latter would be somewhat close to the correct tuning, it would lead to an octave that is about $31\%$ sharp.
TL;DR
Why is compound interest calculated with \ref{comp-int} and not the simpler and (I argue) more mathematically correct \ref{alt}?
 A: One advantage of the current reporting method is that it allows for comparability across timelines.
Banks offer time deposits for periods of 1 day, 1 week, 1 month, 3 months, 6 months, 1 year and 5 years.
We could report different rates such that each would report the total return over each holding period, and the calculation would be what the OP describes.
But, it is not easy to compare rates across time periods.  Under this scheme, the overnight rate might be $0.0086\%$ the one week rate $0.0615\%,$ and the three month rate $0.800\%.$
Instead we say the overnight rate is $3.096\%$ the one week rate is $3.198\%$ and the three month rate is $3.20\%.$
The reported interest rates are simple interest rates for short periods.  Your expected total return, if you rolled over these time deposits for a year would be higher than that due to compounding.  The appropriate compounding frequency would be in line with the length of the time deposit.
Longer-term loans and time deposits are reported on a semi-annual or annual compounding frequency.
Debts are calculated in a similar way.  If you don't pay off your credit card for a month you will be charged a monthly financing fee.  The rate is quoted on an annualized basis, and if you didn't pay your bill for a year, (or charge more debts on that card) at the end of a year your bill would be $(1+\frac {r}{12})^{12}$ times your starting debt.
A: Your suggested formula behaves identically to ordinary compound interest in the case $t$ is an integer multiple of $t_0$ (with $n=1,$ but that’s irrelevant.) Thus there is really no question here: banks banks do not in fact need to calculate your account balance continuously. Instead, they calculate it every $t_0$, where $t_0$ may be a month, a quarter, a year, or some more exotic amount of time.
As for the apparent overpaying if $t_0$ is not one year but $r$ is reported over one year, in fact the financial industry has a solution for that: report APY, the equivalent non-compounding interest rate, rather than $r$, which is called the APR. Of course the difference is usually exquisitely small if you’re saving, though it may be nontrivial if you’re borrowing! So no matter what, the bank always wins.
A: No, that's not how banks compute interest payments rates per my knowledge. Suppose an interest rate $r_{T}$ is offered at maturity $T$. If a client wants $n$ infra-periodic payments that will result in the same $(1+r_{T})$ at time $T$, the computation of the rate $r_{T,n}$ is given by the equation $(1+r_{T,n})^n=(1+r_{T})$ which implies $r_{T,n}=(1+r_{T})^{1/n}-1$. Furthermore, the continuous time cash value in $t \in [0,T]$ of the periodic payment account $A_n(t)$ is given by $A_n(t)=(1+r_{T,n})^{\lfloor nt/T\rfloor}$ obviously if no withdrawals occur.
However, if you compute $(1+r_T)^{t/T}$ you rightly obtain a fair value of the account. That is, the value is given by the current cash held $A_n(t)$ plus the accrued interest $c(t)$ of the next payment which can be given by $c(t)=(1+r_T)^{t/T}-A_n(t)$.

A: There is a practical fact that we really should take into account:
banks typically set their own interest rates.
And different banks can choose different rates.
So let's say Bank $1$ offers the interest rate $r_1$ with traditional compound interest compounded $n$ times per period $t_0$. The balance in the account after time $t$ is
$$
A_1(t) = P\left(1 + \frac{r_1}{n}\right)^{{nt}/{t_0}}. \tag1
$$
And let's say Bank $2$ offers the interest rate $r_2$ with interest accrued according to your scheme,
$$
A_2(t) = P(1 + r_2)^{{t}/{t_0}}.
$$
This implies that $A_1(t) = P\alpha^t$ where
$$ \alpha = \left(1 + \frac{r_1}{n}\right)^{{n}/{t_0}}, $$
whereas $A_2(t) = P\beta^t$ where
$$ \beta = (1 + r_2)^{{1}/{t_0}}. $$
So both banks are compounding the depositors' interest in the same way:
exponential growth.
There are some possible questions about details such as what happens if a depositor at Bank $1$ withdraws their balance at, say, the time $t = 2.5 t_0/n.$
(Do they get only as much money as if they had withdrawn it at $t = 2 t_0/n$?)
And we could ask how Bank $2$ rounds the exponent $t/t_0$ when they compute interest.
But these details just amount to small local "wiggles" in a growth curve that is essentially the same shape as any other exponential growth curve.
Of course which bank pays out more at the end of a period $t_0$ depends on whether $\alpha$ or $\beta$ is larger. If the two banks use the same nominal interest rates, that is, if $r_1 = r_2,$ then $\alpha$ will be slightly larger than $\beta$.
But if Bank $1$ doesn't want to pay more than Bank $2,$ all it has to do is to observe the rate $r_2$ that Bank $2$ has offered, and then set
$$ r_1 = n(1 + r_2)^{1/n} - 1. $$
This will ensure that $\alpha = \beta$, and then Bank $1$ will not be "overpaying" its depositors.
In short, the entire argument in the question is just about two different ways to specify a parameter (and "interest rate") that describes the exponential growth of deposits in an account.
And you may see both versions of the "interest rate" in a bank's advertisements, because while the bank is required by law to specify the interest rate in the traditional way, the second way allows them to advertise a higher numerical percentage in order to lure people to deposit money in the bank.

A historical note: there was, at least in New York in the 1970s, an exception to the observation in the first paragraph, namely, consumer banks were legally prohibited from offering an interest rate above a certain amount ($5.25\%$ if I remember correctly). But interest rates on loans were much, much higher at that time,  and banks would have liked to be able to offer higher rates to compete for deposits that they could then lend out. Since they could not offer higher rates, instead they all offered $5.25\%$ with shorter and shorter compounding periods, with at least some banks advertising that their interest was compounded continuously. That is, their account balance as a function of time was
$$ A(t) = Pe^{0.0525t}, $$
as pure a form of exponential growth as you can get.
