Proposed solution: without changing anything else, just put Somersault and Axe Kick into PvE. Let Somersault give a Chakra stack and Axe Kick a GL stack. Make the recast timers 120 seconds. They don't have to be strong, they just need to be there, and need to be available at level 70 for any level-synced content for MNK to be usable. There's no development cost for these moves since they already have animations and effects working in-engine. They need only be migrated into PvE.
It needs the chakra stack one as well in addition to the GL one because the new meditative brotherhood trait upgrade still doesn't generate chakra stacks fast enough in a lot of cases to make Enlightenment feel worthwhile over The Forbidden Chakra. There's more to say about various high risks of wasting all 5 chakra on Enlightenment when dealing with trash mobs but that's another subject. I would say that Enlightenment's chakra cost should be reduced so it doesn't require a full set of chakra stacks, or its damage output is relative to how many chakra stacks you can spend, to allow for some more interesting meter management gameplay, but that's more of a wish-list thing. As for just single-target content, in a caster comp Meditative Brotherhood still isn't enough from the two tanks alone for tFC to be up reliably as an oGCD substitute, so having one of the oGCDs give a chakra would be useful.
I would argue that MNK lost more than was put back, looking at my hotbars, so I wouldn't think the removal of anything would be necessary, but it really needs both oGCDs as one doesn't feel enough. In the event something does need to be removed, I'd say scrap Anatman since it's too slow and interrupts gameplay. Give them a high recast so they aren't abused and then just call it a day.
If only one of these moves can be added, then allow for SSS to instantly max out GL stacks instead of just duration, provided that there's at least one stack, and this would justify the recast penalty more and make its usage more viable. But it really does need two more oGCDs IMHO and they need to provide some sort of recovery utility for both meters.
Overall MNK got a 4th GL stack but winds up feeling slower anyway and has too many forced slowdowns without enough to do in those windows. One step forward, two giant leaps back, and it became more boring to play.