Google mesh

No U-verse TV Receiver should connect to your Mesh.  If they are, then you have a problem that 3 did, not that 1 did not.  If you have AT&T TV, and not U-verse TV, then you posted in the wrong forum and we'll need to move you and see if someone can answer your question with regards to AT&T TV.

What mesh system works with U-Verse? 

Depends on what you mean by "works with U-verse".

If you mean, "what mesh system can you use with AT&T Internet (nee U-verse Internet)": pretty much any.

If you mean, "what mesh system can I connect my AT&T U-verse Wireless TV Receivers to": the answer is that there is not one.

I mean what mesh system can be used and will it improve the signal strength and/or speed? Or would a wi-fi extender do the same thing?

A mesh system is a router/access point with extenders with a slick interface.

An extender allows you to use some of your Wi-Fi bandwidth to provide coverage to an area that isn't getting good coverage.  Ideally, if you can run a cable to the area where you're getting bad coverage, you would give your extender "wired backhaul" which really makes in an access point.

The AT&T Gateway's native Wi-Fi isn't all that great, so I'd recommend a router or a mesh system to improve your Internet experience.  But if you have U-verse TV, leave the TV receivers connected just like they are.  If you have U-verse Wireless TV Receivers with signal quality issues, your options are limited.

@bjUverseUser 

U-verse these days refers only to TV service.   If you’re asking about generic WiFi, then we need to move you to the Internet forum section.   

I do agree a personal router for WiFi is a better solution than what we get with the gateway.