I don't mean to be pedantic, but BLAST is not a suite of alignment programs. Lots of people, myself included, use BLAST for alignments, but that is not its purpose. In fact, BLAST is not a good aligner once sequence identity drops below 40% or so.
BLAST is a suite of sequence comparison and scoring programs. It says so on its web page.
BLAST finds regions of similarity between biological sequences. The program compares nucleotide or protein sequences to sequence databases and calculates the statistical significance.
The most common use case is “I have an unknown sequence, what organism is it from, or what does it look most like?” In my experience, particularly if you’re a wet lab biologist.
You say those uses are very specific, but that kind of goes without saying - blast is a specific tool for a specific job, but I think you’re vastly underestimating it when you say “it only seems useful to very few people”. It’s probably the most used bioinformatics program of all time. All of those tasks are just an extension of “what does sequence A look like?”.