I reached out to the author of the "Ready... Fight!" supplement for advice on PC builds and how Brawl (Striking) vs. Brawl (Grappling) is supposed to work in various types of engagements. In a nutshell, he suggests that Striking is better versus Minions, while Grappling is potentially better against Rivals and Nemeses ("boss" characters). Either build is serviceable, but there will be a difference in what you can do best. Here's what he said:
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So if it's me? And I have a grappling heavy PC build, and I'm in an encounter with minions? I'm just going to spam takedown attacks for damage, and probably spending advantage to keep my own feet (think steven segal when he's in a fight with 6 guys throwing them all over the place). I won't bother spending an action to improve my position for a takedown, because if the minion group is large, it's probably hard to make that check.
But what you are noticing is correct. A grappling focused build is going to be better than the striking focused build vs rivals and nemeses (the ability to bank crit scores is a huge advantage against such targets versus what strikers do), and somewhat less effective against hordes of minions. If I was building a grappler for the kind of campaign you're describing, I'd absolutely take a punch or elbow attack or something, and a rank in striking, so if I do a takedown and don't finish an enemy, I can finish them with a punch or elbow (though the plain old unarmed attack can work in that instance as well). There is also the ground and pound talent, which is for grapplers that use their grappling to get in a better position for punching.
If you're going to have this sort of PC, I'd just re-read the rules on prone targets, too. Just like in the real world, there is a lot of danger for grapplers outside one on one fights.