Pretty much every supermarket near me sells prepaid phones. And yes, they work exactly as you described. You buy minutes or data to add to your account, and then it sits in your account until you use it. Before the memory shortage price gouging BS, I could go get a burner phone for like $75 at my local Walmart or Target. No clue what they’d cost now, but they’re likely still pretty cheap as far as phones are concerned. The phones won’t have great hardware, but they’ll at least be able to run Android and can do all the things that modern smartphones can do… Albeit much slower than a flagship model.

As far as privacy goes, that will largely depend on your local laws. Drug dealers have historically used prepaid phones to avoid being tracked, (buy a phone card with cash, so you never have to use a debit or credit card to pay,) so governments started passing Know Your Customer laws that require the phone companies to verify and log your ID (tying it to your prepaid phone number, the same way a regular phone bill is tied to you) before they can sell to you. That way when the cops subpoena the phone company to see who owns a specific phone number, the company can actually tell them.