Modernize RECAP codebase

[mlissner]

In #333, we will soon be upgrading to manifest v3. Before we do that, I have this theory that it's a good time to do some refactoring to clean things up.

Let's use this issue to list the modernization steps we should take before we bring a bunch of baggage to v3.

Maybe these are worth fixing later, but a couple things that come to mind include:

• Callback hell. Man, we use a lot of these. Should we stop?
• jQuery. Maybe it's time to not use it anymore? I think we still do, right?
• Using exportInstance and importInstance goes against convention and causes issues #237
• I think we have a branch/fork somewhere that convert it to vue or something. Do we want something like this?

(Of course, if the current version is already fine, we can just close this issue. I'm happy to discuss, but I want to at least have these discussions and decide to carry the baggage or fix it before v3.)