For various reasons, they're mostly used in closed-loop systems these days (think laundromats, transit systems etc.), but historically there were open-loop deployments in many European countries, and in some countries, stored-value POS payments are still very popular, e.g. in Japan.
It's a real shame that the entire world moved to online-only. Sure, it's much easier and there's less opportunity for various kinds of fraud as a result, but in terms of availability during outages or cyber attacks, it was a big unforced step backwards.
Indeed, there used to be things like "Moneo". The problem is that banks never trusted really these systems, so you were limited to, say, 50E of stored value. Also for some reason in Europe the readers of such cards have never been great, I guess because most devices were built on the cheap, so even if the transaction is offline and supposedly fast, you would have to wiggle your cards all around most readers for 2s until it's picked up.
In Hong Kong there is the Octopus card, which started as a closed loop subway card, but ended up being so loved that now you can pay litterally anything with it. It can store up to $500, and you can set it up to automatically top-up to $500 more per day linked to your bank account. Also accepting payments from octopus cards is very easy, you don't need a physical device and small businesses can just have the customer card tapped on their phone with a merchant app.
There were single-use chip cards that you bought and kept in your wallet. Plain eeprom.
I remember I built a reader for them as a teenager...
It also wouldn't work as you describe, as the terminal at the point of entry doesn't know how much to charge you since it doesn't know where your journey ends.
Actual chipcards don't bill you at the end of the month either -- they reload a fixed amount through direct debit (which takes a few days) the moment your balance crosses zero. If the direct debit isn't setup for a card (because it's not a personalized card) or the debit was rejected, the card is blocked.
For business chipcards cards it works somewhat the way you described.
I'm not trying to find a gotcha, but wonder how that works with paying by phone nfc and then by card?
