Filling in details from other papers, is it good or bad?

Let's call the other paper [A].

First of all, you might like to check whether a proof already appears in some other paper or book, [B]. You can search MathSciNet or other databases for papers which cite [A]. If so then you can cite both ("The following proposition was stated in [A]; see [B] for a proof.")

Otherwise, whether to give a proof is at your discretion. If you think the reader would find it helpful to see the proof, and it doesn't distract from the main purpose of your paper, and it isn't excessively long, then sure, you can include it. People often use phrasing like:

We make use of the following proposition, which is stated in [A]. Since [A] does not include a proof, we give one here.

Another option is to put the proof in an appendix, or to write it up as a separate note which you post on arXiv or something.

The Perelman case is about something different - people disagreeing over whether the original paper actually solved the problem, and how much of the credit for the result is due to those who filled in whatever gaps there may have been. In this case, you describe the proof of the proposition as "not particularly difficult", so it's likely that the authors of [A] did in fact know how to prove it, and so would most of their readers who took the time to do it. It's reasonable to give all the credit for this proposition to the authors of [A]. There's nothing wrong with you including a proof, and as long as you don't try to take credit for the result, no controversy will result.

I don't think the situation is at all comparable with the news article you linked. You're not claiming that you should get or share the credit for this proposition, you're not saying that the cited work lacks critical steps that you're fixing. (I also don't think that the proposition in question is a century-old, world-famous conjecture.)

If the proof is really "not particularly difficult", then I see no problem either way: cite the proposition, and then either write your own proof or leave it at that.

But you shouldn't worry about any kind of backlash if you decide to write your own proof. People rewrite proofs all the time for a variety of reasons (you want to match it to your notations, your hypotheses are slightly different, you want to reuse the steps and ideas...) You would have to write this in a rather obnoxious way to encounter any repercussions.