They have a nervous system and DO feel pain, afaik the extent of pain they feel is not really known. A vet had to amputate a limb of an Axolotl I keep as a pet because he got into a fight with another one and he did not even finch.

Why would it be painless?

If the chances of dying after loosing a limb aren't very high it just doesn't make sense to develop (extreme) pain from an evolutionary perspective.

Yeah but regrowing limbs arrived after, it's built on top of their tetrapod ancestry. So it's entirely possible that pain is still here as a leftover from when it was a very critical condition.

The chance of me dying from placing my finger on the stove is 0%, but I assure you the pain is severe.

You can't regrow your finger, and having working fingers would've been useful for survival for nearly 100% of your ancestors (the ones that had digits, that is).

There are lizards who lose their tails painlessly as a way to escape predators. If they felt pain in that situation, they might die, so it would've been selected out over time.