When I used to do full group installs, with new cables, we would always encorage the owner to bring it back in after a few hundred miles for a free adjustment to correct for cable issues. I can understand if they didn't/don't offer such a thing, but would hope that they would oblige, but I/we don't know your shop. If it does require a derailer hanger adjustment and they know they already did that in your build, you might be on the hook for some part of the labor cost. If They didn't check it or don't know if they checked it, it's time to find a new shop or start doing your own work.

You say you've already done some barrel adjustment and it's not helping. I wouldn't go fiddling with the limit screws if you intend to take the bike back to the shop and expect free service. But if the rear derailer is correctly set up, before you started messing with the barrel adjuster, it will be bent inward, and should have had a hard time falling into the highest gear, not the 23. Even if the cable housing compressed (cable stretch), the high limit screw would still be in effect. At any rate, at this point of frustration is where I detach the cable, detach the derailer, check my der hanger, reattach der, set high limit with no cable, attach cable, set tension.

As for the front derailer, it's hard to say without looking at it. But it doesn't look good to me that nothing shifted well from day one. Also, triples suck