The paint bucket tool is a very coarse and primitive tool, one of the original Photoshop tools from the ‘90s. It’s basically Microsoft Paint. Take it for what it is.
What it does is combine two other tools and roll them into one. It’s the Magic Wand (also an original tool), and then it automatically fills the selection with the foreground color. That’s it; no refinement, no tweaking. One shot.
There are so many better ways to make selections now it can’t even be compared. Use those, and when you’re happy with the selection, alt+delete to fill with foreground.
Even better, don’t do it directly on the image layer. Do it with masked additional layers so you can refine as you go.
The thing about legacy Photoshop functions is that as they grow old and obsolete, they aren’t removed. They’re still sitting there because there’s always someone still using it. The better ones are just added. The paint bucket is at least 20 years out of date. Find something better.