Is it ever worth researching mysticism and founding an early religion to get to monarchy quicker?
No. Not unless you have like 4 wine tiles in your capital BFC and another 2 close by or something like that.
HR is useful, but no milestone tech like CS, Currency, HBR or construction etc..
In general the best reason to go the mysticism line is to priesthood for the Oracle.
The next best reason is the need for monuments.
The very last reason is to find an early religion. This is hardly ever worth it, especially on higher difficulties where it can often be detrimental, but there are some somewhat cheesy or weird cornercases in which it might prove a good move. I mention it here, but really it occurs so seldom that you might be better of ignoring it for now.
I'm tempted to make an analogy to bloodletting. Bloodletting was a very detrimental medical practice that was performed for centuries. It was badly used in 99% of the cases. As a heuristic, don't bloodlet is probably a solid one. However, there still are some diseases in which bloodletting is part of a modern scientific medical cure.
Founding your own Hindu/Buddhism is kinda like that. Generally founding one of them is a 'sub-optimal play', and just as I would advice a medieval physician to not ever blood-let again, so would I advise a player wishing to improve to not found early again.
So research them if you go Oracle, research mysticism if you are in need of monuments, otherwise don't bother and trade for them.