Yeah, I don't think there's really any issue with holding Schwab funds in Vanguard or Fidelity funds at Schwab or anything like that, other than that additional purchases come with a fee. ETFs are a non-issue in that regard.
I only hold "cash" in one of my Schwab accounts, and it's all in money market funds (SWVXX). I started taking SEPP from that one in 2018 -- can stop next April when I turn 59.5 -- and I want to keep a cushion of a couple years of withdrawals in cash in case I need to take distributions when the market tanks. That I'd been getting ~5% for over a year made it easier, of course.
But once I no longer need to take SEPP withdrawals, I plan to shift my focus to Roth conversions. And in general, while I have accounts with all of the "big three", if there's going to be an account in the future that needs to have some cash in it, it ain't going to be at Schwab.