Why do we prefer to extend presheaves on a basis to the whole space via limits, rather than colimits? $\DeclareMathOperator*{\colim}{\mathrm{colim}}$Let $X$ be a topological space, $\mathcal{B}$ be a basis (stable under finite intersections) of $X$, and $\mathscr{F}:\mathrm{Op}(\mathcal{B})\to\mathrm{Set}$ be a presheaf on $\mathcal{B}$. Then we can extend $\mathscr{F}$ to all of $\mathrm{Op}(X)$ by defining
$$
\mathscr{F}(U):=\lim_{\substack{B\in\mathcal{B}\\B\subset U}}\mathscr{F}(B).
$$
From a more conceptual standpoint, we are defining $\mathscr{F}:\mathrm{Op}(X)\to\mathrm{Set}$ to be the right Kan extension of $\mathscr{F}:\mathrm{Op}(\mathcal{B})\to\mathrm{Set}$ along the embedding $\iota:\mathrm{Op}(\mathcal{B})\to\mathrm{Op}(X)$. By categorical nonsense, this satisfies $\mathrm{Ran}_\iota\mathscr{F}\circ\iota\cong\mathscr{F}$, which is to say that the extension of $\mathscr{F}$ to all of $X$ agrees with its values on the basis $\mathcal{B}$.
Are there circumstances in which it is useful to instead take
$$
\mathscr{F}(U):=\colim_{\substack{B\in\mathcal{B}\\B\subset U}}\mathscr{F}(B),
$$
defining the extension of $\mathscr{F}$ from $\mathrm{Op}(\mathcal{B})$ to $\mathrm{Op}(X)$ to be the left Kan extension of $\mathscr{F}:\mathrm{Op}(\mathcal{B})\to\mathrm{Set}$ along $\iota:\mathrm{Op}(\mathcal{B})\to\mathrm{Op}(X)$? Of course, colimits are usually much harder to compute than limits, but from the point of view of getting an extension of $\mathscr{F}$ from $\mathcal{B}$ to $X$, this should be just as good: we have $\mathrm{Lan}_\iota\mathscr{F}\circ\iota\cong\mathscr{F}$ by abstract nonsense again, and thus the extension of $\mathscr{F}$ via colimits from $\mathcal{B}$ to all of $X$ agrees with the values of $\mathscr{F}$ on the basis.
As a side, second, question on the possible usefulness of this construction, recall that if we start with a sheaf $\mathscr{F}$ on $\mathcal{B}$ and extend it to all of $X$ via limits, we get a sheaf again. Is the extension of a sheaf $\mathscr{F}$ on $\mathcal{B}$ to $X$ via colimits still a sheaf?
Finally, a third and last question: We could in particular apply this to the definition of an affine scheme, taking the structure sheaf of $\mathrm{Spec}\,R$ to be defined as usual on the distinguished basis, but extending it via colimits to all of $\mathrm{Spec}\,R$. Are there any interesting (good or bad) consequences of doing this in scheme theory?
 A: Think of the sheaf of continuous real-valued functions on a topological space $X$. An element in the product you describe is a bunch of such continuous functions, each defined on a small open piece of $U$. In other words, it attempts to define a function locally. If these different local definitions agree on overlaps, then this is a valid way to define a single continuous function on all of $U$, and uniquely so. In other words, you recover the left-hand side of the expression you wrote with the limit.
Now, if you turn to look at the colimit expression, then on the right-hand side you again consider local descriptions of functions. But, the colimit construction now considers two such local description to be the same as soon as they agree on an overlap. This is certainly not a valid way to describe a single continuous function on $U$. Every continuous function on $U$ will provide you with an element on the right-hand side, but many very different such continuous functions on $U$ will be sent this way to the same element on the right-hand side. So, the colimit expression certainly does not recover the value of the sheaf on $U$.
This should indicate the answer to your question is: this is not a fruitful avenue to explore.
