You can day trade on a cash account as long as you have enough cash to cover all your positions for the time it takes them to settle (T+2)
Consider a scenario where you have a cash account with say 100 dollars and you buy 1 share of stock X for 100 dollars. That's it, you can no longer trade for two days. Because trades take two days to complete, if you have a cash account you need to wait that full two days before you technically own the shares and can sell it. Until that two day elapses, your account has 0 dollars in it.
This is known as free riding:
https://en.wikipedia.org/wiki/Free_riding_(stock_market)
Now if you have 100 dollars in your account and you buy stock A for 30 dollars, then your account will have 70 dollars left that you can use to trade with, but this is an incredibly inefficient and impractical way of trading.
My understanding is that if you had $100 and you purchased stock for $100 and it went up in few hours to $120 and you decided to sell it, so you can purchase something else. You can't you have to wait 2 days.
Of course if you have enough money you will have buffer to account for that, but it makes it harder to do day trading when everything is delayed by 2 days.
The "you can't part" here is what's not clear. My understanding of non-margin trading is that you would be able to sell for $120. What you would not be able to do is then purchase something else with that $120 until T+2 when it settles and the money is in your account again. You technically don't have that $120 until the settlement.
If you can't sell the stock that you bought on the same day that you bought it, then by definition you can not day trade.
Deposit $1000, wait 3 days, buy and sell $100 of the same stock in the same day. You still have $900 to trade with.
It may be “less efficient”, but it also reduces your risk, and risk management is fundamentally what successful trading is about.
Some people don’t want to depend on the bank for margin or leverage, and are happy to trade with cash only. Increased efficiency brings increased risk and decreased ability to deal with short term shocks.