I have done years of research on this topic and found a website that someone else made which is very useful. I am not going to get into a halachic debate on this site but you can go to kohenlevi.org
The MA writes that people in Europe were not makpid on treating kohanim as kohanim bec they do not have yichus. Rav Yitzchak Berkowitz says there is still a big machloches about this and it is not pashut at all. I have a friend for instance who was always a yisroel but is now a kohen after his cousin was nifter and they found a document implying he was a kohen so now he tells everyone he is a kohen, gets first alliya and so on. Rav Moshe though says you need edim mamish and what he had is only a ksav. There are BT all over the world who mistakenly do this and they have children and so on.
Rav Moshe even said you cannot use a tombstone to say one is a kohen.
Even families who claim to have yichus is not enough according to many poskim. So you can quote the Rambam who was nifter 809 years ago but there is a big machlochet about kohanim today. I have asked numerous poskim about this personally.