To what degree is anonymity possible with online payments?

[Kosher Dill]

Alternatively, you can rent a SIM online using Monero

Do tell.

 

[Liber Pater]

Do tell.

Smscodes.io, virtualsim.net and SMSTitan (.onion only) are all examples. I would not necessarily recommend the last of the three, however, as it is buggy, credits expire, and customer service takes a long time to respond. Its main advantage is being an onion site and not even requiring email.
There are more, but these are the 3 I could think of off the top of my head.
Your options expand a lot more if you are willing to convert some of your Monero and pay with Bitcoin Cash, Bitcoin, Litecoin, etc. "Chain hopping" with XMR like this is still secure (if you do it right) even if the coin being used to make the exchange is not a privacy coin, as the XMR transaction(s) will obscure the original origin of the funds you are using.
Personally, I prefer to use services that allow me to pay with XMR directly (as it is more convenient and involves fewer transaction-related costs) but others may feel differently.