However, back in the days of NiCad batteries, some formulations could cope with really fast charges - and in fact would behave better and last longer after a very fast charge than they would after a slow charge.

We're talking about shoving 20 amps into a C cell for five minutes here. You needed to carefully monitor the cell temperature - the charging process wouldn't generate much heat until the cell was nearly full, at which point nearly all the energy going in would convert to heat. 20 amps would heat a full C cell up very quickly indeed.

As far as I've been able to gather (and I have no real special knowledge here and could be way off), the main problem with quick charging is that it heats the batteries a lot, since charging is not a 100% efficient process and any waste energy turns into heat. Heat is bad for batteries, so it could affect their life. However, Teslas have active temperature management for their batteries (both heating and cooling, as required) which should mitigate that.