This the part of my morning where I’m scraping through all the tear downs of 3DS style devices to see if there’s an X-Ray of the cartridge reader in any of them just for sake of clarity. But no dice.
I have not specifically worked on one of these, so you can take what I say with a grain of salt, but after having a look at the outsides and using my morning hyper-focus, my inkling is that this is likely a product of two mechanisms. One to keep the spring from ejecting the cartridge and one to keep the cartridge from falling out. I suspect they may actually be the same part within the cartridge reader, but separate things. I suspect your issue is with the former. That’s the thing that gives the satisfying click when you’ve inserted the cartridge far enough to engage the spring lock.
Update: Realized I never finished my thought. It might be difficult to fix if it is the “Click” that’s not engaging. Because the bits that actually handle this will be inside the cartridge reader, and unlikely to be accessible. I might see if I have something similar I can rip apart just to see. I have DSi XL and a DS lite in my apartment somewhere I can compare.