For some of the older questions, some of the appropriate tags are missing. If tags are added, the questions are bumped to the front page.
Is this seen as undesirable, and should adding tags to old questions be avoided for this reason?
It's not undesirable, but at the same time, just exercise some restraint. When adding a tag to an old question, try to also improve the question in other ways too. It's less-than-ideal to retag more than ~5 old questions at a time since they'll drown out new questions at the top of the home page.
One similar note, mods can merge or do a tag rename without bumping questions, so if the re-tagging falls into that category, by all means, post a meta question with a list of questions.
Incidentally, there's also a feature request on meta.stackoverflow along these lines (prevent bumping on a simple retag).
I have been re-tagging questions on the site. Mainly because every question (following the recommendations) should have 3-5 tags. In this way, the tags will help to find relevant questions...that's what they are for! Yes, they will be bumped up in activity, yet that seems fine since there is a "Newest" view if you want to see the latest questions.
In my view, this site is still evolving and tagging all questions is about curating what is already here as well as setting a "best practice" for future questions. By setting a strong standard of quality, the site will get better search results both from inside and outside.
The purpose of tags is to make it easy to find relevant questions and detect duplicates. As well as finding groups of similar questions. But in order to perform their function, all questions have to be properly tagged.