Where I live if you fall and injure yourself on an established hiking/climbing route, in season, you will be rescued for free (regular ambulance rides are free anyway).
If you climb or hike off track, out of season you might be charged for a helicopter ride or mountain rescue. As you can imagine it's not cheap.
That should be on case in all of Europe (in Germany and Italy for sure). It's covered but your insurance as long as there is an emergency. So you can't just use the ambulance as a taxi, then you would be billed the whole amount.
Not entirely in the Netherlands. There you have a yearly mandatory "eigen risico" of €385 (or more if you want a discount on health insurance). The first costs you make, you have to pay out of pocket. Only after you've exceeded the €385 in a year will the insurance start covering the costs.
This seems quite reasonable to me, and if you are doing the more risky activities, then presumably you'd get some insurance to take the edge off the rescue charges.
If I take an unregistered, uncertified, uninsured vehicle to a place that has no roads and I get myself in a huge pickle out there, you can bet your ass I will be billed for the rescue. And probably fined on top of it for fuckin up the environment in the process.
Also as a hiker, I have a satellite comms device in case my ass needs rescued. And I have SAR insurance, because my ass cannot afford the bill.