Well, they did. In separate ways.
The Ottomans were in the beginning tolerant towards other religions, accepting Christians and Jews. it's not for nothing the expression "better the Turk's turban than Pope's mitra".
This is a very interesting expression; interesting for many reasons, but chiefly because of how misleading it is.
The man who
allegedly said it (Loukas Notaras) was actually a supporter of the Catholic-Orthodox church union; in other words, a huge fan of the "Pope's mitre". And, incidentally, he and his entire family were murdered by the Sultan.
Which suggests the whole expression was a misunderstanding or a slander. In any case: that it was definitely exaggerated and probably straight-up false.
That's not to say that there were
no Byzantines who really did think like that. They definitely existed, but there is nothing to suggest that they were in the majority (or even that they were a large minority). In practice, a
lot of Orthodox (both rulers and self-managed communities) chose to ally with Catholic powers against the Ottomans.
It would also be fair to note that few or none of the Catholic powers operating in the Balkans were really of a "convert or die" sort. They also had at least some religious flexibility.
And Byzantium itself was reasonably tolerant back in the day. IIRC Constantinople had a Muslim community and a mosque at least until 1204.