I’ll split this up because I have a character limit for some reason… My apologies! - Edit Found a way to solve this. Sorry for the remaining double posts!
I think SMN in itself is on the right track but should rely on their summons more. I feel that a lot of its potential is untouched which is saddening which is why I'll try to make some unsolicited suggestions to expand the current SMN instead of completely revamping it. I understand it not making sense to have every SMN summon a full-blown Ifrit and I don't mind egis.
This is why I think the DoTs suit the class. It's an indirect playing style whereas the source of your direct hits could be your pet. But as many pointed out the pets seem to be the add-on. They are not the main source while you are in the background doing your DoT work and keeping them alive. They are the “background noise” or occasional DoT enhancer.
(I also think its no crime if XIV gives some of the FF legacy a different spin if done well and explores different routes to some extends.)
I wish they would put some more spotlight on the summons like having them learn at least one or two more important skills and not stick with four forever and maybe have them grow a little as you progress (improved aether management lorewise), and/or maybe slightly elaborating on the design every ten levels or so (comparable to armour that gets fancier and fancier to indicate how badass and highlevel you are.) Also the fact that some are almost universally preferred over others is not good imo. I know this is incredibly hard to balance but I would love to for egis to have more actual situational use or a bit more freedom to choose between summons for one situation in a "different but equal(ly effective)" type of way.
And while I really like dreadworm stance->Deathflare I’m afraid that choosing the ">summoning a beast/stance for one strong blast and then it'll disappear<in different versions as main source for DPS" route, if one were to revamp SMN, might ultimately turn it into “just” a strong caster with fancy animation and story. Its uniqueness might rather stem from the lore/visuals and not so much from mechanics. I think splitting damage by micro-managing pets and inducing indirect damage as back-up is quite unique and personally I enjoy this a lot.
One last idea – I’ll pretend to be joking while being at least half serious. 
Another idea would be the Shiva route – turn into a beast yourself for a short amount of time. This might also help to work around the "no more egis" issue without sacrificing the identity of the summoner.
It sounds wacky but I think it could be unique and effective if well-planned. As you have limited aether you’d not be as strong, big and longlasting as a real primal ofc but would have a very short time frame to cause condensed higher damage.
Originally this was inspired by the manticore orb in DD. However the best comparison I have for this would be when you play charizard in supersmash and have your mega evo as smash attack for a limited time. Of course this would need a lot more balancing (than the manticore for example) but having one (or two, depending on the future level caps) “ultimate commands” that let you channel a primal within yourself for ~10 s and you get 2 or 3 skills+stats that make sense for them to cause a damage peak, then slap a long(er) cd on it – I would so play that…Also, since it happens in the actual story it would be in line with FF14's idea of summoning.
Maybe glamour could also be used much later on to customize your final form.