You absolutely need to trade on the same security intraday many times in order to generate a profit as a day trader.
Buying a security, then buying a different security, then selling a third security tomorrow -- that's not day trading, that is just the normal flow of equity investing.
Unless you have a lot of money so you can be invested into many different names, you won't be able to do this very many times before you run into the limit, at which point, the rational person will open an account at a different brokerage instead.
The utility of 'zero commission trades' rapidly declines as you have more capital. If you have >$25k to put into a brokerage, paying $1-2 commission per trade (e.g. Interactive Brokers commissions), or even $5-7 per trade (most other retail brokers) is definitely worth the flexibility of being able to change your mind and reversing that trade without the risk of being locked for 90 days.
As for the addendum, sales in cash accounts are settled over several business days and you can't trade on those funds, so again there is no day trading there unless you're very significantly overcapitalized.