At its core,

[Poopadoop]

Does the "feeling" being preceded by other chemical processes in a brain mean that the feeling is wrong?

If those brain states precede and predict the feeling of making a decision, then I would hazard to say 'yes the feeling is wrong'; your feeling of having made a decision is a phenomenological artefact of the activity in your brain, nothing more.

We constantly make decisions without any sort of rumination. Are these decisions 'free' in the sense that we could have done something else? It seems hard to agree with that idea. And I'm not speaking solely about reflexive or automatic actions - just think of the next conversation you have with a friend - how much of what either of you say do you actually think about before you say it? Not much if you're like me and my friends, you both just blather on.

Surely you've done things where you've had no idea wtf fuck compelled you to do them.

I'm not sitting here questioning if I should have chosen different weather for today. My brain is not simply asserting the feeling that I've made a choice over arbitrary things in my life.

That just shows you understand what is possible and what isn't. Other people do sit around thinking they can affect all sorts of things they can't affect. But what you think you're in control of and whether you have the free will to decide to change it seem orthogonal to me at least.

Does it fit the observed data to posit that a brain makes a decision, and only after said decision is made takes the luxury of "feeling" like it made a decision?

In a sense, yes, but I think you're ascribing too much sophistication to what the brain is doing - it's basically just a bunch of 1s and 0s that are programmed to follow certain patterns - e.g., respond in such a way so that the person turns their eyes and head towards a loud noise, for example.

The real mystery is why we have consciousness at all, since our brain could arguably pull all the right levers without us ever having any sort of phenomenological experience of even existing, never mind feeling, thinking, seeing, being happy or being sad, etc..