Alright now that I've seen the deadly sins cards it's time for me to step in
For his "mind control" to take affect, he needs to hit his targets with arrows. He's not hitting any of them with arrows because of their CoO Haki. Not to mention that even if it did hit, they would only be under his memory swap for a few seconds before they realize the inconsistencies and break free.
Yes, he needs to hit his targets with his arrows in order for it to work, and he show that he can hit several Holy Knights with a few moves, meaning his accuracy is great. Furthermore, I'm afraid you are using observation haki as an ultimate evasive technique that can dodge anything and that is not true. We saw that even Enel with his observation haki of the size of an island still can be hit. Admirals didn't show their observation haki to be not even close to that so they can be hit. You conclusions are based on nothing.
Your argument that even if they can be hit they would be under his memory swap for a few seconds is also based on nothing, since admirals never show any feats on being resistant to mind control and since it's shown that Gowther's skills work pretty good and he can get enough time to mess with Holy Knights. I really don't know why you think Admirals will be an exception of Gowther skill.
And I have read every chapter so far of SDS. He usually doesn't use a weapon for the reasons stated here. Unless you'd consider his broken sword a weapon.
And even when she gave him that sword, he had to have Ban convince him to use or else he wouldn't have.
Since we can consider his being very powerful with a wooden stick (slicing a mountain in half with it) of course we can consider a broken sword to be his weapon. Look at the volume cover one and two, he is holding his standard equipment/broken weapon.
Meliodas slicing a mountain with just a stick
Kingdom of Danafor completely destroyed.
You can also see that his range is great so he doesn't even need to get close to the Admirals.
All it needs here is Gowther and Meliodas to defeat Admirals... and if you have a doubt we can always add immortal Ban and two more sins. There is no way Admirals can't win round one.
Round two is a whole different story...
Alright, I'll dispute your claims in the order that you went
1. Observation haki can most definitely predict the arrows that this Gowther can shoot. For one unless you show a scan that his arrows move at relativistic or light speed, you're going to need to say more than just 'they can definitely hit because they're accurate'. The problem isn't how accurate they are, it's how fast. Luffy can dodge swords being tossed at him at hypersonic speeds out of gears
While there isn't much I can argue about those memory things, if the arrows have to hit for it to work, then I doubt he is going to implant it like you think. You're also going to have to show me a speed feat from anyf them that puts them in admiral speed range
Also we've been over this feat, it's high town to small city level. I'm sure with a sword it's probably city level + but even without intangibility the admirals have island level durability. Receiving an island splitting quake to the face and getting up is quite impressive for Akainu.
Gowther could try and do hand to hand but will find a haki fist to the face from either one of these guys. Or he'll be frozen. Or incinerated. Or lasered. Take your pick. And as for gowther, how high is his durability? Same as the other sins? because if not, a couple good lasers which are very accurate and possibly much faster to the body will finish him. Or a shot to the head, which if any save for Ban gets hit to the head with, will die. And Ban will get frozen.
No, they don't win round one like you think. They have the numbers advantage, but at the moment the Admirals are superior in strength and speed to anyone on their team.