The minhag in Eretz Yisrael is that after the first round, the rules loosen up quite a bit.It is still preferable to call a cohen first, but then a levi should be second, etc… In a case where for example there are two cohanim left and no one else, the second cohen is called up like this:” ya’amod ploni ben ploni af al pi shehu cohen”.
As far as tying up the chazzan and other shenanigans, this utterly despicable “minhag” exists only in chutz la’aretz. It’s shocking that frum Jews would even consider this behavior.