SAG kindly replaced my cart and the replacement arrived yesterday. I've been testing it, and I've concluded my console must be to blame.
I'm still scratching my head on what's going on. My console plays original carts fine, and I've never had any modification or maintenance done on it. Using original PSU. And I'm thinking it could be some kind of memory/bad trace issue that messing with the NeoSD Pro specifically. The issue can manifest on original carts, but usually only after I hit the reset button. Something about resetting my console, or some function on the flash cart that resembles resetting, somehow freaks out my console. When I first power a game on, it plays fine.
I put in my original SamSho 3, and it boots up and plays just fine when I use the power switch. When I load a (known-good) SS3 ROM on the NeoSD Pro, I get graphical glitches. Sometimes I can see the menu's text glitch while the game is flashing. It looks something like in
this image where columns are repeated or staggered somehow.
I likely need to get some TLC on my system, and from all I can gather, the NeoSD Pro is not faulty after all. And it speaks wonders to SAG's customer service that they exchanged mine when it could just be my old ass system was the problem after all.