Other than relying on "side-chains", is there any way that Bitcoin could solve "micro transactions"?
The main challenge is that any transaction that is transacted on the main chain will have to be stored on the blockchain in perpetuity, and processed by all full validating Bitcoin nodes. To my knowledge, all cryptocurrencies have the same property... they just aren't large enough where this is a real problem yet.
The other thing you should consider is that the actual value of a transaction to the Bitcoin network as a whole is irrelevant. A 1 satoshi transaction takes up the same blockspace as a 1 BTC transaction.
Hopefully this explains why small transactions must take place off-chain to achieve the cheap and quick settlement you mention.
The goal of a project like Lightning Network is to have microtransactions take place so they aren't taking up block space and needing to be stored and processed by all Bitcoin nodes forever. It will hopefully become easier to use as time goes on.