I've also recorded Vinyl LP's on my PC and turned them into CD's. If the Vinyl is in good shape you can't tell a difference, you can even edit out skips and scratches and remove a lot of static with the filtering. But if the vinyl is really bad, you can improve the sound but you can't work miracles.
The problem with recording Vinyl on PC is that you need a turntable with a pre-amp built in or buy a pre-amp to put into the signal before the PC. Sound Cards don't have pre-amp like most stereo amplifiers do. Many new turntables have a pre-amp built into them, because turntables are so out of date the new amps don't have inputs or pre-amps for turntables.
It takes a little more time than making a tape, because you have to fiddle with the files when your all done and split up the songs, then filter out the noise and pops, etc. But once you do it and have a CD, then its much easier to make mixed CD's.
Eventually, you'll have to transition over, if you need a new stereo, might as well do it now. Of course, it will probably cost more than finding a used cassette player, and converting your old cassettes. But, this course of action can't be continued forever, eventually you'll have to convert over, and then you'll have to spend all the money all over again.
If your son wants the cassette just because he mixes his own tapes from CD's, and his collection is on CD. Then its a no brainer, its easier to make mixed recordings with a PC and CD Burner than it is with CD to cassettes.
Plus, having a CD player that can play MP3 from the CD-R is like having a juke box. One CD you can put 110 songs on it, you can play hours of music without changing the disc.