Different classes always trade with different tickers. When they trade, of course: Google has also a non-traded B class. (By the way, I don’t think the first statement is correct.)
Prohibited was the wrong word, you're right. Institutional investors that run ETFs have become averse to dual class stocks ($SNAP, $LYFT) because they don't have the same voting power per share that they do with other share class setups. Indexes like S&P 500 have also stopped including listings that are dual-class -- this is changing though, see the articles I've linked below.
- https://www.cii.org/dualclass_stock - https://citywireusa.com/professional-buyer/news/crown-duals-... - https://www.washingtonpost.com/business/dual-class-shares/20...