Is my 802.11n router the problem and if so, what's the best solution?

New Option: Buy your own Wireless Router and use it for Wi-Fi functionality in place of the AT&T Gateway's older Wi-Fi.  You can connect it to the existing Gateway and disable Wi-Fi in the Gateway.  See also 

Configuring IP Passthrough and DMZplus - AT&T Internet Support

So, you're saying our 3rd floor wi-fi is weak because of the Pace Gateway's 802.11n protocol is inferior to newer routers, or because it's a cheap unit and has an inherently weak signal, or... what? You say buy our own. Okay... I am sorry to say, I don't even know if we pay a monthly rental fee for this unit, or not.

  But you're saying just pass the signal through this Pace gateway to our own, new and better router (better because it will use a newer protocol)? And, I can skip buying the range extenders altogether?

AT&T's gateway's have questionable WiFi coverage, especially if you have a 3-story home so follow the above advice and purchase a better router for WiFi. Preferably a mesh WiFi router and place the satellite on the third floor wherever you want better coverage.

ISP-supplied equipment, any ISP, typically does not have the most robust routers/gateways so you're always best to use your own, if you can, and avoid the monthly fee with better equipment as well.

Thanks a bunch. It's just silly how ISPs cut corners thinking customers won't eventually resent them for it.  Following your advice, I just (at 11:59 PM) bought a TP-Link Deco Mesh WiFi System (Deco S4) – it claims up to 5,500 Sq.ft. Coverage, Replaces WiFi Router and Extender, Gigabit Ports, Works with Alexa (who cares), the 3-pack at Amazon was marked down for Prime Day to $113.  I almost never (!) catch those sale prices! I suppose when it arrives I'll learn I missed something (like when we bought our first flat screen. Samsung plasma TV, got it home, and learned it came without the HDMI cable (or was it two) we would need to use it. That meant another half-hour drive each way from Key Largo to the mainland's Best Buy or Walmart, and paying a premium but getting (more or less) instant gratification, or waiting a few days for a delivery from Monoprice or AmazonBasics. I'm sure I waited. Anyway, thanks also for the encouragement to go for the mesh... I was wavering on that vs the old format. But given all the concrete full of steel rods between each floor, I think the three pack mesh system was the way to go if I want reliable wi-fi on our 3rd floor.

By the way, also in the last hour, I read something (for first time) at CNET.com: that most "range extenders" come with an inherent 50% loss of signal speed...

"Signal loss happens because extenders have to do two things at once: receive the signal from the original router -- a process often referred to as back-haul -- and rebroadcast that signal to Wi-Fi clients. Having to do two jobs simultaneously, the extender's efficiency is reduced by 50 percent. This means clients connected to an extender, like your phone or laptop, will have just half the real-world Wi-Fi speed compared to those connected to the original router."

It seems like that's something that should be mentioned more often.

Yes, range extenders rob your home's overall wireless bandwidth and add latency to anything using them.  Whenever possible, range extenders should be used with wired backhaul, which turns them into access points.  Without wired backhaul, they must also be placed where they can obtain a good signal, which may make placement in a multistory home tricky.

You could use powerline bridges to provide wired backhaul if you absolutely cannot run Cat5e or better twisted pair.

I agree with the above. I have my mesh WiFi system hard wired for the back haul. With mine, the satellite has 3 extra ethernet ports so I can hard wire other devices to it. Mine are hard wired with solid copper core, CAT-6 cable (non-CCA/CCS).