$73$ really is the smallest prime $r$ which cannot be represented by means of
\begin{equation*}
\frac{pq+1}{p+q}
\end{equation*}
because as you noticed before,
\begin{equation*}
\frac{pq+1}{p+q}=r
\end{equation*}
\begin{equation*}
pq+1=r\left(p+q\right)
\end{equation*}
\begin{equation*}
pq+1=pr+qr
\end{equation*}
you may add $r^{2}$ on both sides,
\begin{equation*}
pq+1+r^{2}=pr+qr+r^{2}
\end{equation*}
\begin{equation*}
pq-pr-qr+r^{2}=r^{2}-1
\end{equation*}
\begin{equation*}
\left(p-r\right)\left(q-r\right)=\left(r-1\right)\left(r+1\right)
\end{equation*}
is highly composite on right side; anyway making $r=73$
\begin{equation*}
\left(p-73\right)\left(q-73\right)=\left(73-1\right)\left(73+1\right)
\end{equation*}
\begin{equation*}
\left(p-73\right)\left(q-73\right)=72\cdot74
\end{equation*}
\begin{equation*}
\left(p-73\right)\left(q-73\right)=5328
\end{equation*}
so $\left(p-73\right)$ and $\left(q-73\right)$ are divisors of $5328$, which possibly are
\begin{equation*}
\{1,\;2,\;3,\;4,\;6,\;8,\;9,\;12,\;16,\;18,\;24,\;36,\;37,\;48,\;72,\;74,\;111,\;144,\;
\end{equation*}
\begin{equation*}
148,\;222,\;296,\;333,\;444,\;592,\;666,\;888,\;1332,\;1776,\;2664,\;5328\}
\end{equation*}
but the only prime values of $p$ such that $p-73$ is a divisor of $5328$ are
\begin{equation*}
p\in\{79,\;89,\;97,\;109,\;739\}
\end{equation*}
then,
\begin{equation*}
\left(p-73\right)\in\{6,\;16,\;24,\;36,\;666\}
\end{equation*}
with these values, $q$ cannot be prime because,
\begin{equation*}
\left(q-73\right)\in\{888,\;333,\;222,\;148,\;8\}
\end{equation*}
\begin{equation*}
q\in\{961,\;406,\;295,\;221,\;81\}
\end{equation*}
none of them is prime.
On the other hand, here are some representations for primes $r$ up to $97$, except for $73$
\begin{equation*}
\left(p,q,r\right)\in\{\left(3,5,2\right),\left(5,7,3\right),\left(7,17,5\right),\left(11,19,7\right),\left(13,71,11\right),\left(19,41,13\right),\left(29,41,17\right),\left(23,109,19\right),\left(31,89,23\right),\left(31,449,29\right),\left(37,191,31\right),\left(41,379,37\right),\left(43,881,41\right),\left(71,109,43\right),\left(71,139,47\right),\left(59,521,53\right),\left(71,349,59\right),\left(71,433,61\right),\left(89,271,67\right),\left(73,2591,71\right),\left(131,199,79\right),\left(89,1231,83\right),\left(113,419,89\right),\left(109,881,97\right)\}
\end{equation*}
and about the infinity of such numbers which cannot be represented, I strongly believe they are infinite, since I found others like $73$:
\begin{equation*}
\{73,\;107,\;131,\;157,\;173,\;179,\;193,\;227,\;263,\;277,\;283,\;313,\;317,\;331,\;367,\;383,\;389,\;457,\;499,\;503,\;509,\;523,\;557,\;563,\;653,\;673,\;677,\;691,\;761,\;787,\;823,\;829,\;877,\;887,\;947,\;983,\;997,\;\dots\}
\end{equation*}
I tested those with all combinations of primes $p<10^{6}$ and $q<10^{6}$.
