Convexity can be a very important assumption as much of economic analysis is built on working with convex sets, which makes things easier.
Very importantly, convexity of sets allows us to work with the Separating Hyperplane and Supporting Hyperplane theorems, which have applications to many results in economics, as partially discussed here and here.
Here is also a short, concrete example of something for which we need convexity of the budget set:
Suppose you want to prove that Marshallian Demand $x(p,m)$ is single-valued for every (larger than zero) price ($p$) and (non-negative) income ($m$) combination. That is to say, a Marshallian Demand Function exists, since without single-valuedness we would have a Marshallian Demand Correspondence (i.e. we'd have multiple good combinations as part of the optimal demand choice for a single given price and income level), which can be harder to work with. This is a pretty nice result that we (almost) always work with in undergraduate texts. For that you need strictly convex preferences and a convex budget set $B(p,m)$. The proof is as follows:
If two points $x_1$ and $x_2$ are both elements of the demand $x(p,m)$ at given prices $p$ and income $m$, i.e. if $x_1, x_2 \in x(p,m)$, then by definition we must have indifference between the two points, that is $x_1 \sim x_2$. For a single-valued demand function two different points $x_1$ and $x_2$ cannot exist for single given level of prices and income, i.e. they would have to be the same point. In other words $x_1 = x_2$. We prove it by contradiction. Suppose $x_1 \neq x_2$.
For all $\lambda \in (0,1)$, we have $\lambda *x_1 + (1-\lambda) *x_2 \in B(p,m)$ by convexity of $B$. By strict convexity of preferences we have $\lambda *x_1 + (1-\lambda) *x_2 \succ x_1$. However, this contradicts the fact that $x_1 \in x(p,m)$ in the first place. Hence, $x_1 = x_2$.
So, in words, some combination of $x_1$ and $x_2$ is better than just $x_1$ at the given price and income level. But such a combination is also attainable at that given price and income level because of the convexity of the budget set. This is where the budget set convexity assumption is important. Furthermore, if for a given price and income level there is a point (in this case a combination of $x_1$ and $x_2$) that is both strictly better and attainable, then this contradicts the original starting point that $x_1$ is a point in the demand function for the given price and income level.
As a side note, I think it is a bit misleading to talk about producers in your post when talking about budget constraints. If we are looking at a budget constraint, we should be discussing consumers.