Also, let me know what airports in Europe you can operate a for-profit taxi service out of without paying the requisite fees and having the proper licenses.
I'm going to interpret "out of" to mean "picking up passengers from" not "having an office in" and "requisite fees and having the proper licenses" to mean "without paying the special airport fee" but that you may have to be a registered taxi. If you meant something different please ask again more clearly.
In which case I will answer that I don't know of one where you can't. In some such as London, where you pay #50 just to get into a taxi you would be insane to do anything other than have a "mini-cab" (these are registered, but with much lighter requirements than a proper taxi) arranged to come and meet you. You just have to do the arrangement by phone. What that cannot be done is for those taxis to wait in a taxi rank.
This is a good example of a regulation which is done for the good of the customer. In many airports there used to be serious cowboy taxis who would wait for tourists and overcharge them massively by driving around a long long route or simply by having outrageous hidden charges or by various other kinds of fraud. By regulating and ensuring that the taxis that stand outside the airport are known, that doesn't happen much any more. The locals then take the cheaper local taxis which come and pick them up at pre-arranged places so this doesn't cause much overcharging.
The Heathrow's #50 service fee is a perfect example of a failure of democracy. The airport authority has a monopoly on air transport and even so is allowed to get away with doing whatever they want. This is one example of why I said "most of Europe" rather than "in Europe".
I would REALLY want to know. Because, you know, "In most of Europe yes; the regulations are there in order to improve people's lives and especially safety."
This is still largely true. Nothing is perfect and you shouldn't expect it to be. If you have a problem with a regulation then try to get it changed. If the majority of people agree with you that it's a bad rule and you still can't, then start thinking about how to change your political system. If they think it's a good rule but are wrong then start thinking about how to educate them. If you can't do either then you have a problem.