"a) look on the 'what is playing' section of the station's web page [a lot of them do this]"
Not that handy to do when you're on the go, especially if you don't know WHAT station is playing, or even if it is a radio (it could be a dedicated stream personalized for the shop, so no playlist), and then by the time you open up the website and look it up, it could already be on the next song and they keep no history.
b) if it's internet streaming radio, you'll see the ID text displayed [probably]
Unless it's a PRIVATE stream. See (a).
c) who really cares, since RIAA only excretes CRAP these days, with rare exceptions"
What about stations playing older music, say from the 50's through the 80's? If you're going to say this music is crap, either you have a tin ear or you just don't like music, period.
PS. Shazam and the like are actually QUITE good with older music since it tends to be pretty popular. Foreign music may be another matter unless the music provider keeps an international database.