Lost Mode without Internet

So the other day my phone got stolen an the thieves managed to get a hold of my emergency contacts which was then just very confusing to me, since I’ve activated the lost mode and was turned off ever since it got stolen.

However, it was already turned off when I put it in Lost Mode and they’ve probably turned it on with the removal of my SIM card and zero access to a connection.

However, they’ve connected my phone to the internet once and that’s where I got an E-Mail notification that my phone has been now sent into Lost Mode.

It was very clear to me that up until that moment they could’ve accessed my phone without being in Lost Mode and therefore accees my medical ID and my emergency contacts.

Shouldn’t it be possible by now that iPhones could be sent into Lost Mode without it being connected to the internet?