I knew a Protestant convert to traditional Catholicism who spent the rest of his life spreading the message that we should NEVER use Protestant hymns in Catholic Mass because it's improper to God. But even on a purely human level, it's rude to converts because when they hear such things in the setting of Mass, they are struck with wondering why they converted if it's the same content they thought they had left behind.
When a convert hears "A mighty fortress is our God" or "Rock of ages" or "Amazing grace" or even the "doxology" (...For Thine is the kingdom, the power...) appended to the Our Father, he is reminded of all the teachings he learned among Protestants, which he has been trying to forget, and it all comes rushing back like a bad dream.
So out of respect for their journey, at a bear minimum, we must avoid such tunes.