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No, I was completely serious and I really like your friend's idea. Sorry if that came across as sarcasm.
As for twentieth vs twenty-first, I think it's better to keep twentieth, unless you also change the cardinal twenty to twenty-zero, which doesn't seem useful. So nineteenth, twentieth, twenty-second, twenty-twifth, etc.
My apologies for the misunderstanding. I agree that (conditional on doing all this stuff) you wouldn't want "twenty-first" corresponding to 20 even if you had "twenty-second" corresponding to 21.