What about stoat, weasels and ferrets? I imagine one of those kills at least three fold more wildlife and birds than a cat, not counting eggs. Mustelids basically drove the kakapo to extinction.
I'm not a conservation expert, but think it's safe to say that the answer to your question is a bit complicated. We have a national goal to be free of a list of predators including those by 2050 - it seems achievable, but only with a lot of work and political willpower. Stoats for instance can be killed with traps that cats can't get in to, so that's a useful tool in the more accessible areas where we still have pet cats to contend with. One of the more immediate issues in the big picture, is a small but vocal faction that's violently opposed to using a poison that is key in these efforts. To give some flavour, check out this recent and very well-done series: https://www.rnz.co.nz/programmes/fight-for-the-wild
No, we don't. The only requirement is a chip + rabies shot if you are taking your cat, dog or ferret with you across other European countries (per directives 998/2003 resp. 576/2013).
> My ex lives there, has lost her cat, it was captured by animal control and she had to pay a 300€ fine at the animal shelter to get it back.
That is not a fine by the government, it is a charge from the shelter for the trap, medical checkout and accomodation cost.
On the other hand, cats are commonly seen as a lower cost and lower responsibility option, and as a result there are loads of them loose outdoors. I help maintain pest (mainly possum, and rat) traps in a forest at the edge of a small city, our traps include some self-resetting rat traps [1]. This isn't something we've kept close track of, but I'd estimate 1 in 15-20 rats killed overnight in those traps will still be left the morning after, because a cat has cleared the dead rat overnight.
An outright ban on cats is a political non-starter, but if we could at least require registration then pest control folks would have some chance of eliminating feral cats outside of special conservation areas. And, hopefully that change would shift the discourse a bit so a ban could be feasible down the track.
[1] https://goodnature.co.nz/products/a24-rat-stoat