All that is true, but I think the biggest advantage of monk is subtle blow and reduced TP feed with Chi blast(penance).
This is a fantastic point, just highlighting it again. And like Simon just posted, I'd suggest having each player level and gear 2-3 jobs - with enough flexibility to have players covering the important roles of tank, healer/support, and DD - is a great idea.
I'd consider things like how much gear overlap there is between the jobs, and how easy it is to gear the jobs without requiring super high end gear / the job being useful without necessarily requiring tons of different gear sets. For a 3rd job, I'd pay even more attention to something that fits pretty well with the gear that these players would already have or be pursuing on their "main 2". So, my votes would be for something like:
Player 1: PUP MNK... and NIN if you want a 3rd
Reasoning: PUP and MNK share weapons and some good staple gear (Mpaca set, Malignance set). PUP tanking doesn't require a RMEA, MNK still puts out solid DPS with Ambuscade H2H. NIN as a 3rd option is due to it also sharing a lot of endgame-capable gear with MNK PUP (Mpaca Malignance for all 3 jobs, and a lot of other stuff is shared between MNK NIN SAM even if PUP isn't on it).
Roles: PUP is great for situational pet tanking use; MNK is a great DPS that can take some hits and feeds minimal TP to the mob; NIN is pretty flexible as a solid DPS that can tank+DD on many fights where shadows aren't frequently wiped or ignored.
Player 2: SMN WHM... and GEO if you want a 3rd
Reasoning: Lots of shared gear, none of them need a RMEA to function at a high level. SMN JSE Oboro staff is very good; WHM has lots of non-RMEA options to address it's primary role of curing; GEO can just buy a JSE neck and/or Dunna for much stronger buffs than trusts.
Roles: WHM is straightforward, cure the people! GEO is also pretty straightforward as a buff/debuff job.
SMN is where it gets interesting - while the job can be a little complicated to learn, the actual gear requirements aren't excessively high (a JSE staff, even the NQ Ru'Aun abjuration set is a very solid pet set, grab some NQ Su3 gear for Summoning skill for buffs = you've already got your basics sorted) and the job can address several different roles. It can be played quite effectively as
(a) a buffer (I've been using it a lot in Omen card farming duos-trios this past couple weeks!) - Hastega II + Cornelia or a Bard trust to cap haste, Crystal Blessing for TP Bonus, Ifrit's Avatar Favor for Double Attack+, Earthen Armor, etc.
(b) SMN can do some good backup healing/status removal with /WHM or with avatars. For instance, you might be watching your party to help supplement a trust who might not quickly remove status effects, or who might get killed in a fight so you can step in to help toss cures to try to survive.
(c) Can also be more DD-focused with pets, especially fast fights where Astral Conduit avatar damage can really murder stuff (and even if you can't kill in AC window, can knock out a lot of mob HP and work from there). You'd have a very good pet-focused trio with PUP SMN COR.
Player 3: COR PLD... and WAR if you want a 3rd
Reasoning: COR is the only pet buffing job, and has strong melee buffs + ability to add significant DPS for non-pet scenarios. PLD fills a tank role for fights where PUP (or NIN) aren't as viable, and there still are many of those. WAR because it shares a lot of gear with PLD (both are perfectly functional with little more than 5/5 Sakpata set) and can be a very powerful DD without needing any RMEA weapons (lots of very strong WAR-usable Ambuscade and Unity weapons - Naegling that you'll also want for COR and PLD, Loxotic Mace, Montante, etc.)
One note about why I intentionally left THF out of contention... the reason people are suggesting it a bit seems mostly down to Treasure Hunter. But IMO, TH is overrated for content most trios might do. It's totally useless in a lot of currently important content like Ambuscade or for getting Odyssey Gaol gear/RP. And the big factor:
any job can already get TH+4 fairly easily and tag mobs in situations where TH helps (Omen for Swarts, Sheol A-C for Lustreless items, High Tier BCs for drops, etc.). Two all jobs TH+1 accessories are very easily obtained (Chaac Belt from legacy Delve content, Perfect Lucky Egg from a small bit of progress in the Voracious Resurgence missions), leaving only one lucky Oseem Dark Matter TH+2 augment for any job in the game to have the max non-THF Treasure Hunter value of TH+4. The difference between zero TH and TH4 is noticeable, but the difference between TH4 and TH8+ is MUCH more minimal. If you're trying to do a trio that is as flexible as possible, I just don't think THF as a "pretty good DD that has TH" is as valuable of a role as many other choices.
EDIT - some practical party compositions for situations you will realistically run into:
1) PUP SMN COR - for pet focused fights with puppet tanking/DDing, avatars doing damage, COR buffing with pet rolls & potentially doing some safe ranged DPS.
2) MNK SMN COR(or PLD) - for low TP feed setups, with MNK as your DPS/tank using Chi Blast and subtle blow, SMN for Mewing Lullaby and buffs, COR for more buffs. COR could change to PLD if a "real tank" is needed instead of the MNK doing both tank+DD roles.
3) PLD WHM MNK - for the classic tank/healer/DD trio, effective on mobs where you need a real tank that can hold hate off a melee DD (PUP will often struggle here), or getting hate on multiple mobs (a PUP weakness), etc. Mostly for fights where you do want a real tank, but where PUP (or NIN) aren't cutting it, which will happen more than you may realize.
4) WAR NIN and any of GEO/WHM/SMN - parties where you might be moving around and fighting numerous mobs that each die fairly quickly (for example, Sheol A-C or Omen farming). Your DDs are both pretty sturdy and can take a hit but also dish out good damage. You might have the NIN be "first voke" and soak up hits with shadows, but it's not the end of the world if your WAR pulls hate.
5) NIN COR GEO - good setup for blink tanking on single mobs that don't excessively wipe shadows. NIN tanks, holds hate, and adds solid DPS. COR does damage and give buffs (but is less likely to inadvertently pull hate like a WAR). GEO buffs (especially ensuring the NIN has capped haste) and debuffs the mob, plus some backup for cures/status removal, and could even add magic bursts if the NIN COR are skillchaining.
This really shows the elegance of the FFXI party system, and is honestly a lot of the reason I've continued to enjoy the game for so long. There's so much depth, so many ways to approach a situation, and so much interesting interplay between multiple jobs using their strengths to fit in combination with another job's completely different strengths. Even having to think through this "what's a good trio setup" is just such an interesting and complex question, and the fact that there are so many ways to approach that question is precisely why FFXI never seems to get old for me.