It's more like HP shares splitting into HP and Agilent, and your brokerage deciding that it won't track your ownership stake in Agilent and not giving you any further access to obtain your Agilent stock.
Under these circumstances, you'd have serious reservations about passively parking any assets with that particular broker...
That's one possible analogy. Another analogy would be: an unrelated third-party decides to give, to everyone who has HP shares, an equivalent number of shares in a new company. In that case, a broker would be under no obligation to give you these new shares.
Which analogy is the best one depends on your point of view on this debate.
Correct. I have not studied securities law but I don't think Coinbase's users have any recourse claiming their BCC and Coinbase decides their stance is that it was always theirs as they held the relevant private keys.
Actually... with many brokerages, you can still obtain actual paper stock certificates -- it's just that the overhead and fees to do that make it highly unattractive.
Furthermore: It is possible to directly transfer stocks from one brokerage account (similar to a wallet) to another at a different brokerage without having to liquidate assets.
Under these circumstances, you'd have serious reservations about passively parking any assets with that particular broker...