Home WiFi Dropping Internet, Router Problem?

I'm hoping that someone can help me here at least get an idea of what my next steps should be; I don't mind buying something new but I want to buy the right thing and not end up having to buy unnecessary equipment.

Here the problem that I'm having:

We've had AT&T High Speed DSL for the last 7.5 years and we've had the same modem/router combo that whole time; I think it's a Westell Versalink 327w. Anyway, it's old but it's worked fine for us until just recently.

Last week we bought a Smart TV but we set it up pretty far away from the router so it wouldn't reliably connect to our WiFi. On the advice of some people I spoke to I bought a WiFi repeater (this one: Belkin Dual-Band Wireless Range Extender) and set it up to cover that area.

The repeater puts out a good, reliable signal that the tv has no problem connecting to and, when it's working the TV apps (primarily Netflix) work great. The issue is that we lose our internet, not our WiFi, that signal stays, but we lose internet connectivity very often. I definitely notice it more often when we are streaming Netflix. My phone, laptop, TV everything still sees and can connect to the WiFi, but the WiFi can no longer get on the internet. When this happens I go reset the modem/router by turning it off and then back on and once it starts back up it works fine for a while (until this happens again).

Is this a problem with my old router/modem? Can it not handle this added stress of the repeater and the tv?

Here's what we normally connect to the internet through the WiFi: the repeater and tv I mentioned above, my wife's laptop, my wife's and my smart phones, occasionally my laptop, a desktop computer that is almost never on, and a PS4 that gets used a couple of times a day but very rarely for online gaming.

Can I buy a new router and connect it to that Westell modem/router combo if I turn off the WiFi signal on the Westell? Can I even connect another router to a modem/router combo or will I need to get a stand alone modem if I want to use a new router? Do I need a new modem anyway since what I have is so old or will a new router fix the issue even with the old modem? Any other recommendations?

I've called AT&T customer service but they can't tell me what's going on; they can't see any problems with our equipment or signal from their end. They've tried to sell me a new modem/router but I don't want to buy that unless I know it'll fix the problem (also, I wonder if I might not be able to get something better if I bought something somewhere else).

Thanks in advance for any guidance you guys can give.