As Snaggles points out, a full BiS Monk will have a primal weapon, so STR, DEX, and AGI will be capped regardless. If you really cared about having 255 dex for proccing Avatar initially, you could just ask for a DEX buff or Avatar, then click off the extra buffs once you get your first proc.
You are correct that any race can cap their STA with BiS gear and a Shaman STA buff.
However, Monks don't have a good non combat stat to dump into if you are going for Min/Max. I don't believe Monks get Divine Intervention cast on them enough to worry about it. Warriors have CHA for Divine Intervention, hybrids have INT/WIS, etc.
So for Monks STA is still probably the best choice because it allows you to reduce the number of buffs you need by 1, and it allows you to swap gear around for resistances. For example, a BiS monk would have
https://wiki.project1999.com/Robe_of_the_Azure_Sky as a chest swap for resistances. So you would lose 10 STA right there. You would also have
https://wiki.project1999.com/Shawl_of_Protection , which would reduce your STA by a further 15. So wearing Robe + Shawl means you are down 25 STA. Having +30 all resistances over your current setup is generally going to be better than 150 worn HP, so honestly I would probably just wear Robe + Shawl all the time and forego getting the Tunic and Pauldrons.