There is no DIRECT business reason to “hire more people”.
Ah, but I think, in your example, you are looking at things backwards (the market forces aren’t giving an incentive to hire more, but rather not to fire everyone), and also, you are looking at an example on the micro level that doesn’t necessarily translate to the macro level.
Even so, you cut off your example right before it would obviously have the same result I predicted, because this:
“So, eventually, Business B will go under, and Business A will capture a portion of whatever customer base B had remaining.”
...is not a sustainable situation that would result in a new equilibrium. The only reason Business A can hope to capture more of a dwindling customer base is because Business B hired more workers to keep the customer base viable. Since Business B has shut down, those workers can’t be counted in the customer base anymore, so there isn’t much for Business A to capture, and therefore Business A will also have to close its doors shortly as well.
Business A would see labor as an expense. Two companies A and B exist there must be some market for widgets. Company A will continue to exist. Company A can easily adjust operating cost.