There are basically 2 ways
1- to find an individual on where his/her line first commenced and (BCA test)
2- where one's admixture changes every 6 months based on new members joining. (Gedmatch type of tests)
There are also the "weird" system of getting/using only 5 testers for data for each ethnicity ( dodecad) and using these mimimum testers to claim most European results instead of continuing to gain more testers.
or
The PF ( Ftdna) system where they have a "centre of gravity" system where one is pinpointed to a map of Europe based on ethnic areas...example... for me ....62% Tuscan and 38% Occadian/french....so lets move 38% of the mileage distance from Tuscany towards Britain and mark it...that's the pinpointing.
or
The 23andme system of going back only 500 years
Sile, are all these "ways" you describe ways to show or prove someone is European? I know something of your ancestry. Of course you're European...why would you doubt it? I really don't understand. I can trace most of my ancestors back to the middle of the 16th century, and some even further back...all Italian. I have no doubt that the the vast majority were in Italy for a thousand years, at least, before that. My ancestors tilled this land, built its great architectural wonders, engaged in its commerce, participated in and benefited from the Renaissance, which, in my opinion, defined what it means to be a member of western civilization. I'm Italian, and European. What else could I be?
EVERYONE in Europe has ancestors who arrived from somewhere else. EVERYONE. It just depends when they got here. I'm personally not interested in drawing a line in time and saying everyone whose ancestors got here before X date is European, and everyone who arrived later, is not.
For those who are, I would just say that generally human variation is clinal. You can't contain it within political boundaries, or even within geographical boundaries. As the whole series of articles by the Reich group at Harvard shows, the "European" populations were formed by an admixture of a North Eurasian group, with similarities to the ancestors of the Amerindians, and a west Eurasian group probably radiating out of the greater Near East somewhere, some of whom arrived in the Mesolithic, for example, some from the Neolithic, perhaps some from the Bronze Age. All Europeans show admixture from those two groups, albeit in different proportions, and forming some general clines within Europe. That's the way it is. It's not as neat for Europeans as it is for the Han, who form a more distinct pole, or the Yoruba, on the other hand, who form another pole.
As for the "ways" you listed, they are, of course, all flawed.
To base one's identity on one y line in your ancestry, which represents only 2% of your genes, seems rather silly to me, I must say. Just as an example, I carry mt dna U2e. The latest research I've seen said U2 arose in the Middle East after the Out of Africa migration. Much of it went to India, but some headed into Europe. The U2 at Kostenki is tens of thousands of years old. It has been found from Andronovo and other steppe cultures all the way to Basque country, and everywhere in between. So, which do I pick? Am I, for example, supposed to feel like a steppe dwelling pastoralist? I assure you that I don't.
The other option is autosomally testing for "admixture", at least admixture on the level where it can be done or accessed through such things as the Dodecad calculators. You seem to have some issues with it, but I think it's a much better option than chasing after one "Y" line to see when your particular mutation hit the shores of Europe.
I, like you, am no fan of the FTDNA analysis. I'm likewise not a fan of Dr. Mcdonald's program. He created both programs, you know, and although he has changed his own algorithm a bit, it is still the same general method.
Of the other calculators available, I personally have the most faith in those done by Dienekes, for the simple reason that the populations, except for his own "Dodecad" members, are all publicly available, and he published his methodology, so anyone can duplicate his analysis on their own, and ensure that it is honestly done, given of course that they have the computer skills that are necessary. (BTW, there are far more than 5 participants in most of those studies; there are many samples available for Tuscans for example. The proof that you don't need all that many samples to get pretty accurate results can be seen from the results for Ashkenazim, for example. You can compare the results based on the Behar samples, and the Dodecad volunteers, and the results are almost identical. Or, take me, for example, my ancestry comes almost entirely from the corridor which runs from Parma to La Spezia. My dodecad results are without exception almost exactly midway between the scores for Bergamo and Toscana. )
That's not to say that I think these tools can't be improved. That's precisely why new populations keep getting added; it's to make them more accurate. Also, this analysis only goes back so far, not as far as the admixture event(s) proposed by the Reich group for example. The clusters are also somewhat ambiguous, and change from run to run because they are experiments done to try to figure out the peopling of West Eurasia. They weren't done for the "consumers".
As for 23andme, I don't actually think it only goes back 500 years. I think that statement was put out there mainly to cover themselves because at this point it can't be proved exactly how far back it does reach, but that's a different discussion, and this is already way too long.