You can argue wether bitcoin should act as cash or not — different people have different use cases. Some use it as a long-term store of value, others use it only for speculation.
I don’t know the numbers to back this up, but my gut feeling is that bitcoin isn’t used all that much as a payment mechanism, online or in physical stores. It’s an interesting ideal but in practice I’m not sure it happens that much.
Anyways, solutions like Lightning will make it possible to use it as cash, if people would like that. It’s definitely similar to your cheque analogy — but settlement times are faster (~60 minutes from the moment you decide you want to settle), and you have the option to never settle and still be sure your “cheque” is valid.