There are many halachot that we don’t keep today because we don’t have a bais hamikdash. Why then do the halachot of a Cohen still remain? E.g., not being able to be in contact with tumas meis, not marrying a divorcee? Why was chazal metaken that these halachos still apply if the Cohen can’t do the avodah today, and if most people can’t truly prove that they are a Cohen?
The kedusha of a kohein is intrinsic to his being. He is born with it and takes it to the grave. These halachos are not takanos, but rather Biblically ordained. The fact that he can’t serve in the Bais Hamikdash does not change his status, and is ultimately a technicality. Many kohanim have clear tradition as to their kehuna, and although it may not be provable, this is certainly sufficient to act accordingly with all the halachos of a kohen.