at the time that the problems were solved?

Just to focus the discussion on that point, I'll guess No.
Perhaps someone can show a practical problem in architecture where finding the height of a arch or something like that requires a cubic or quintic. That would still leave open the question of whether architects solved the problem that way or whether they used scale models. There is also the question of what is considered a practical application. For example, Cardano was (among other things) an astrologer. He no doubt considered an application of mathematics to astrology to be practical. Do we count it as such?
I think the original post is asking whether the solution of polynomial equations was the focus of intense interest due to the need to solve such problems using the technology of the era when these solutions were developed. The history of math books I read (once upon a time) never mentioned any such pressing need as motivation  but perhaps those books were written by pure mathematicians.