You are right; Rollover is a process, and not an account type; the result is a Traditional IRA. There is no such thing as a 'Rollover IRA Account'.
Rolling a 401(k) over to a Traditional IRA makes sense if a) you have to, because you leave the employer the 401(k) is with; b) because you Traditional IRA is cheaper or more flexible or in other ways 'better' for you, or c) if your next step is a backdoor rollover to a Roth IRA.
Most of the time, it doesn't make sense, because employer 401(k) are often better and cheaper. Of course, for the investment company where you roll it too, it makes a lot of sense, because they get your money, so they recommend it. But that's only good for them, not for you.
Of course you can roll into an existing account, if you want to roll. Making a new account has no advantage.
I cannot imagine any IRA custodian wouldn't take rollovers; they would shoot themselves in the foot by that. What can happen - and you should consider this - that your IRA only accepts cash, and does not allow to transfer the shares you have in the 401(k). That means you have to sell and then re-buy, and you might lose a lot in fees there.