Being of a practical mind, I would suggest whatever paper you select should have a paper base, which excludes the great generic genre of luster papers. RC papers look great but are a total PITA in the handling and mounting department, and they outgas for a long time after printing. I feel paper based prints are just easier to deal with and have no significant outgas problem.
That may narrow it down. Epson Cold and Hot Press papers make exquisite matte prints, as does BC Elegance Velvet. Elegance can also be successfully coated with normal canvas acrylics and presented without glazing, unlike most other papers. Cold and Elegance have slight surface texture which really doesn't get in the way. And BC Optica One makes positively dazzling smooth surface matte prints thanks to an abundance of OBA's.
Of course there are many others. You won't get as much gamut from the matte papers as from the glossies, but then it's not really how much gamut you have, it's how you use it. Sorta, depends on your subjects. I'm not current on baryta papers, but I bet you could find a happy choice with one of those. Lots of color gamut with those barytas, you can make full use of all those ink cartridges.
As for profiles, you just have to make them or have them made, end of story. I regard my profiling system as basic equipment. Have always seen improvements in the profiles that I make in house versus manufacturers profiles. There's a difference between "in the ballpark" and "spot on" profiles, and "spot on" usually requires a profile made for a specific, single printer. IMHO. And that is true both for my present Canon and my many earlier Epsons.
And me too, I use a lot of Epson Exhibition Canvas Gloss and it looks awesome on the wall. Same in-your-face impact as a glossy paper, no problem. But as with all glossy surfaces, it requires mounting to solve reflection issues that come from subtle surface ripple seen in wrapped or hinge mounted prints
FWIW, my BC Crystalline and EECG profiles are almost interchangeable, extremely close. Same is true for Epson Cold Pres / BC Elegance Velvet.