1) it definitely is more busy then any other DPS job with managing your pet, DoTs, Aethertrail stacks, pet buffs for fights it takes longer to ramp up your DPS compared to other DPS like Drgs and Blms but their sustain is really good. A lot of summoners just blow their load and don't know how to stagger their Aethertrails they see the 30 seconds and go "crap I have to use it now!" when they could: Aetherflow, wait for fester to come back from CD and then pop Dreadwyrm, Tri Disaster, Fester and finish your Ruin III Spam. A lot of summoners do not see that because combined with Contagion those DoTs tick in the mid 1500-1600s which is HUGE.
2) Smn is very rewarding and can compete with DPS like monks, drgs, nins etc it's all on the player/gear myself screwing around in Gordias Normal for a friend to get their relic on Gordias floor 1 I finished at 1776 DPS and i'm only ilvl 228 with a Nidhogg weapon. People want dynamic summoners instead of a DoT class which I understand but because of this fantasy class design that they want they're turned off of Smn from the start and never give it a chance.
3) The question comes down to: Do you like DoT dmg? Do you like controlling a pet? Do you want to strive and improve to be a better Smn? If you said yes to all of this and practice you'll surprise yourself learning the rotations and DoT priorities make the difference between a good and bad Smn. Deathflare combined with Foes can reach some insane dmg upwards of 12k and with all the crit stacking on a Smn and your pet with DoTs is actually kind of scary just how much dmg a Smn can really do, it's not as big number heavy as a blm but it's there in the DoTs and Ruin III + Deathflare crits before Ruin III used to be 120 potency now it's 220 which is huge.
4) All I can really say from my perspective as a Smn in A5S and A6S is that my dmg isn't mediocre I main Drk but this and Drg are my off DPS classes and in A5S in my current gear and knowing the fight in and out I can end at 1400-1500 DPS and sustain over 1700. To me Smns aren't a fifth wheel I enjoy it, does great DPS, and brings battle rez, Eye for an Eye and Virus which are great defensive abilities. Give it another chance and do the Stone, Sea, Sky dummies and if you can clear them and feel positive about it try EX Trials and see if you like it in an 8 man raid setting or even Midas normal if you want to do that as well.
Sorry for the huge post I hope you give summoner another chance it's the most underplayed job in the game and a lot struggle with it but with all DPS classes practicing and getting comfortable is the most important if you can DPS and do mechanics just by muscle memory and conquer that you'll be in the clear.