Not overkill, but not really practical either. There is a mushroom shaped flexible blade for that, but I don't use it for such situations.
This is my favourite tool for that job, but I cannot find it in local stores anymore:
LinkI also have this thing, but I don't want to waste the supersharp thin blades on tile, I use it in other situation, like with glass, countertop/backsplash etc.
linkSomeone posted a scraper with a bunch of blue replaceable plastic blades a while back, I haven't tried such a thing yet.
Often I end up using a replacement stanley blade with my fingers, but that's no fun (the regular scraper holder for those doesn't come as close to the corner as I want)