In your opinion, what is your personal go to when it comes to combatting severe rain for footholds in the warmer months?

For what?

Cage trapping?
Foothold?
Beaver?
Exclusion?

Postpone a day or two. No ladders or roof work in the rain unless I absolutely have to. Will set cages in the rain for ground animals. I set foot traps about .0003 % of the time, so not an issue. There are exceptions to every rule, I take them as they come.

footholds, my bad for not mentioning

In the spring/summer when you're nuisance/ADC trapping, and the traps are already in the ground with rain coming, let 'em ride. The great thing about nuisance work is that you're not skinning water logged, muddy animals. The traps will perform slower in muddy conditions but will still catch critters. Once it has dried out, re-make the sets.

I'd never use a foothold for nuisance work. The moment a neighbor carrying a smartphone and some sort of social media account looks over a fence and sees a sad looking animal in a foothold you're in for a world of PR headaches. Set a cage.

100%. Cage traps only.

Only time i might use DPs or something would be in an extremely rural setting, even then it’s not likely most rural people will just trap the critters themselves. I’ve heard of guys using body grip traps inside attics and other enclosed spaces but I’m even hesitant about that for fear of a “bad catch” where the animal is flopping around making all sorts of god awful noise. Customers don’t want that

I would guess the original poster operates in a similar environment or even more rural area than I do in the Atlanta market.

Footholds aren't the issue, it's the discernment or when and where to use them.

If neighbors with cameras are a possibility then you take the appropriate steps to cut off that neighbor or don't set.

Pretty much this.

I try to make sets that carry away water and remain functional through rain. I use a method very similar to the Zaggers dream set that leaves a bowl or pocket under the trap as a sump for water. I also make my sets on a slight slope as well. Done right, I usually don't even need to do a remake once it dries.
In our clay, particularly in summer, that wet dry cycle can get some spots hard as concrete when dry so that has rendered traps inoperable.

x2 with an experience

When setting footholds in a spot that will likely get very wet, I take a long stake and drive it in the bottom of the hole as a sump, and use poly fill under the pan. Winter time I place the trap in a plastic bag.