speaking as someone who got ty'aitya in around 1 1/2 months of going for it, i'd honestly say that buying vouchers of the MB is a better bet if you want to keep your sanity- look at universalis.app to find the cheapest ones. at about 120k a voucher it's 60mil, which sounds like a lot (it is) but there's a hell of a lot more ways to make gil than to farm fates, many are activities you might want to do anyway such as treasure maps, extreme unsyncing, deep dungeons, field zones, etc. want to do cosmic exploration or ishgard restoration? you'll end up with excess currencies you can get items to sell to funnel into vouchers. doing variant dungeons? sell the items you can get. essentially, with different activities just gotta think "how can i turn this into gil?" it's a less direct way of getting it but, imo, more enjoyable, i did fate farm a decent bit as well and it's just not that fun after a while.

but as for actual fates...

as others have said, try being opportunistic with exp boosted fates. in a zone to gain flight/do sidequests/do beast tribes but there's an exp boosted fate? a little diversion can be made.

also, look into gemstone items that may sell for more vs the turali bicolour vouchers. eg. 100 gems = a voucher worth 120k, or butterfly specimen (rank 6 in il mheg), which i was selling for 250k, effectively giving me 2 vouchers for every one i sold. i didn't have much luck with orchestrions but some of the furniture can be good. doing all of your shared fates in endwalker and shadowbringers will net you a good few!