Schweinsteiger was considered finished by every one I know who watches football. He'd been injury plagued for a couple of years, wasn't getting in Bayern's team and he arrived with a dad belly and grey hair, he was completely finished! Why would Bayern have let go an absolute legend of club and country for next to nothing if they thought he still had it?
Dani Alves was the best right back in the world last year. He was the best player in both legs of the Champions League semi final. He wasn't great in the final, but it doesn't mean he's finished, he was playing against the best left back in the world, and against the best team in the world.
There might not be many examples of 34 year olds coming to the PL and being brilliant, but I can't think of one that came off the back of such a good season as Dani Alves. The closest would probably be Ibrahimovic, but he stopped running a few years ago and played in a sub standard league. Alves hasn't shown any decline physically and has just played in the CL final, so he's clearly playing at a very high level.
Probably before your time, but City bought a 30 year old full back called Tony Book and he went on to be our most successful ever captain and won the footballer of the year. Let's hope Alves can have a similar impact.