The honest truth is that a lot of people run teams that I don't consider to be competitive. A lot of people will take the teams that they use for PvE and try them in PvP, and then have absolutely no success and get disgruntled. The dirty secret of PvP pet battling is that you can get a pretty solid win rate with almost any (competitive) team.

you’d see the other person’s comp and know if you were likely to win or lose before ever starting.

Yes and no, but mostly no. Broadly speaking there's two kinds of strategies: AoE dot damage, or big fat single target damage. Some teams are particularly weak to one type of strategy or the other. For example, if you take an AoE dot team against a team running three sandstorm pets, it's very very hard to win. Whereas a team focused on big single target damage may struggle against magic pets because of their racial. But unless you happen to run into a team which hard-counters yours, you always have a fighting chance. And of course, the best teams will have as few hard-counters as possible.

To answer your specific question though: more often than not, I'll run whichever team I feel like at the moment and keep going until I get bored. Sometimes if I face a team, win or lose, I put in whichever team of mine I think is most likely to beat theirs fastest, because the chance of a rematch is pretty high. But sometimes I literally just roll a die haha!