I believe the Asakusa 24Kaikan opened in 1912, not as a gay sauna but rather as a place for cross-dressers to meet in secret.
Public bathing and gay saunas are in no way related, as PeterRS has indicated. Public bathing is all about social interaction, often based on the Japanese belief that, although their society is hierarchical and everyone knows his or her place, everyone is equal once they take their clothes off, as there is no indication of their status.
My friends and I were frequent visitors to the sento (public bath), not because we didn't have our own baths but because we loved to chat with others.
I, of course, had a separate interest!
And lots of guys had an interest in the foreigner - I often found myself surrounded by guys wanting to see my cock. But I doubt they were gay; just curious.