I think your problem is that you're looking at "end-game" and only seeing "Chi Blast." There's a difference between the two.
First of all, even when you are "just" using Chi Blast to down the gods (and this really makes up less than 1% of your time unless you log in solely for this purpose), you're in the very unique position to not only do serious damage but also to be vital backup for your tanks. Those Monks who don't wuss out and sub Ninja to gods are there with their White Mage sub. That gives you Cure III, Curaga II, Divine Seal, Raise, and Erase. Put yourself to use while you spend 3 minutes Boosting. If you don't like either option when dealing with the gods, level up your Thief sub (just because under-levelled subs suck) and join the SATA rotation. If you're really devoted to using melee, I'd recommend that you level up your Staff skill, since Full Swing (or, hell, Retribution) is going to do more for you than Dragon Kick.
Your role at HNMs varies greatly, both as per your LS and per the mob being fought. Let's take the Three Kings, first of all, since that's what most people think of when you say "HNM." At Adamantoise, if you sub Ninja you can very easily solo the local mobs, you can even tank two at once if you have a Bard, Red Mage, or White Mage watching you. Additionally, though you'd be hard-pressed to convince your LS of this, you should be able to tank Adamantoise. I'm hedging that in very tentative tones, though, as I'm not completely sure of it. For Aspidochelone, you're really only good for handling adds and maybe using Chi Blast.
For Behemoth... hell, Monk can solo Behemoth (or used to be able to, not sure on the rage timer here) and duo or trio him painlessly, the hard part is getting claim. For King Behemoth, it's the tough-shit rule because he's kited, so it's Chi Blast or nothing, since you're not a Ranger.
At Fafnir, Chi Blast is an essential part of dealing damage but, frankly, Warriors and other jobs melee Fafnir and Nidhogg to death, so there's really no reason a Monk couldn't do the same. Fafnir and Nidhogg, however, are where a Monk/White Mage shines as a support job. Unlike that Tarutaru White Mage in his Aquarian abjurations and Serket Ring, you can drop DS + Curaga II and tank the wyrm for a few rounds while your main tanks can happily go nuts rebuilding their enmity. It beats the pants off watching your mages die when you can put on an Earth Staff, maybe a Jelly Ring, and laugh thanks to your high VIT and HP.
The CoP wyrms resist Chi Blast. They don't resist Sneak Attack very well. Speak to Jepedo about some of the nasty shit he's pulled on Taimat using Full Swing. You don't even need Sneak Attack to do it, either, just spend a few minutes Boosting to 999 Attack, put on your Accuracy gear, and hope you don't miss (one of the many situations I'd probably recommend using Sole Sushi). If Jepedo, a Tarutaru no less, can put up nearly 2000 damage with Full Swing on Tiamat, there's no reason you should be pumping out 600 damage Chi Blasts. Same deal goes for Jormungund, though I imagine you could just as easily tank the Ice Elemental adds as Monk/Ninja. Snolls I am unsure about. And for Vrtra, you are an add-killing machine, after all, it's what we XP on day in and day out.
Even still, that's not a whole lot of "end-game." Want a challenge? Solo Serket. It's still possible, you just can't dally around with him going berserk after 30 minutes. Better yet, camp and solo Alkyoneus, a fast-hitting, hard-hitting, level 75 Warrior NM that fights with Hand-to-Hand. Not exciting enough? Go tank Capricious Cassie, Roc, or Simurgh.
Still not enough? Well, to fight those gods you're so fond of being bitter about, you need pop items. Tanking Aura Pots is painless, they can even be solo'd if you're smart about it (with the added bonus that Monks don't aggro them), and your LS will love you for getting Ro'Maeve Water. You could just as easily tank Olla or Steam Cleaner. In fact, show up to virtually any non-god event prepared to tank, with the possible exception of Faust (although, I imagine you could pull off some nasty things if you went about it like a Paladin and used VIT/HP gear and an Earth Staff). Go XP/Merit on Aura Statues, that was always my favorite thing to do and you'll be camping Ullikummi in the process.
If all that bores you, there are a lot of non-end-game activities still open to you. Personally, I solo'd and duo'd my way through the Windurst Missions, solo until 6-2 and duo until 9-2 (I tanked 9-2, by the way). Go help people with Genkai II and enjoy beating the living hell out of the Boreal Hound. Do like I did, start a collection of obscure Rare/Ex items, work your way up to killing NMs that are hard to find and harder to kill. How many people own a Power Staff, hmm? It's a great way to outfit your lower-level jobs. Back when I was just a wee Monk, I camped Cargo Crab Colin for a Nadrs sword, which I used to own mobs with on my Thief as well as impress people ("Where did you get that sword?

"... it looks like a Greed Scimitar). Complete the Zilart Missions, even. Having watched Avesta's video of solo'ing ZM17, I'm firmly convinced that a Monk should be able to duo it with a mage. Or you could take up a craft seriously. Or camp money NMs, just for the fun of it or for the profit or both.
If all else fails, there are a dozen other jobs available for you to level. Solo'ing your AF quests for another job was fun for me, I wish I'd gotten Ninja higher so I could have done them all. Honestly, the end-game content in this game is somewhat lacking, but resigning yourself to hating it all just because one aspect of it differs greatly from your usual day-to-day as a Monk is just silly.