Granted I only travel with hand luggage, I never had any issues with cancelled or delayed flighs when booking separate tickets. Even so they are relatvely cheap. It is even cheaper if you book your return flight ticket separately.
> Granted I only travel with hand luggage
That makes an enormous difference, since you usually can do your check-in online a day or more before departure.
I just had a flight where I had purchased my connections on separate tickets. When I arrived at the baggage drop-off, it was closed and no staff from the airline was anywhere. So I had to bring my big suitcase along through security and then the airline staff at the gate was friendly enough to check it in for me for free.
But if you're bringing anything in your checked luggage that security doesn't like, you will have to just leave that item at the airport. You might also have to pay a high fee to bring it to the gate instead of checking it, depending on airline. I had purchased my ticket directly from the airline, so maybe they treated me better because of it?
You need to be more clear about what you meant by saying "never had issues".
The point is there might be issues because the second flight has no obligation to refund you if you're late (but they might still do out of generosity).
It's not like it's gonna happen all the time. So having data points saying that didn't happen is kinda pointless.