Would my router cause us to have wi-fi connectivity and be connected but repeatedly have no internet? This is a nightly occurrence.
Absolutely possible, but not for sure. Assuming your Century Link internet is DSL I would generically say the most likely points of failure are -
A. Router - This could be its basic networking function, the wireless delivery, or possibly using poor DNS servers in the DHCP settings (sorry for getting in to some slightly technical abbreviations)
B. Modem - The gateway from your router to your internet service provider's data lines. ISPs are content to have these to sit un-upgraded for a long time because they tend to just do their thing. They do age or become technologically outdated, but not as rapidly as a router.
Note - it's not uncommon to have a device that is both the modem and router. In general these combo units perform worse than having a separate router and modem. Especially with wireless performance. The manufacturers of the combo units don't seem to put the needed emphasis on keeping up with the very fast changing wireless standards and quality.
C. End devices - wired computers, laptops, smartphones, TVs, tablets, etc. If they are junk or are bogged down by other things then performance will suffer.
D. Connection strength - This is a can of worms and has 2 main areas.
First, the connection from the modem to the ISP which is traditionally via physical, buried lines. Not in your control and ISPs don't like to have to troubleshoot this.
Second, the connections of your equipment to the modem. Basically, if you've got poor wiring somewhere along the line you will suffer.
If connection issues are the culprit for speed issues then finding it can be a challenge. And it's tough for an ISP support tech to hep diagnose because they can only "see" up to their modem. What goes on in your house is out of their hands. They want to avoid dispatching someone out to you because it costs them a lot more than having someone chat with you on the phone. A good, patient support tech can help you do some testing to diagnose what is going on. But if you're going through a big ISP's residential support portal then this will be unlikely.
A big part of this connection in your case is the wireless performance. If you have an old router in the corner of your basement and are trying to stream to a smart TV across the house it can be painful because of distance and potential materials reflecting or absorbing the signal along the way.
Keep in mind that wireless is not magic. It's set of different radio frequencies that obey the same laws of physics as other sound waves. The frequencies are just much higher and inaudible.
I'm starting to ramble a bit now so I'll wrap it up with some abbreviated advice for those having issues and ideally have access to a laptop -
1. Do an internet speed test or two while on wifi in a few different places. Speedtest.net is fine. I like sourceforge.net/speedtest quite a bit too. Write down the results noting your location.
2. Plug in to an open network port on your router and repeat the test. Write down the results.
3. If you have a separate modem then there will be a network cable from the modem to the router. Unplug it from the router and plug it in to your laptop. You should get as clean as a connection as possible like this. Repeat speed test and write down the results.
4. Reboot everything excluding the modem and repeat steps 1-3
5. Reboot everything including and starting with the modem (wait a few minutes for it to finish connecting back to the ISP) and repeat steps 1-3
If you have big swings in your results before rebooting the modem then the problem is likely on your side. Upgrade the router and try again.
If things improve with a reboot of the modem then get on the line with your ISP to get help looking for connection issues. Stay patient and sound confident (but not accusatory) while you tell them your service experience and results from testing. If they actually acknowledge that you've put in the work and aren't someone calling to complain about speed and want magic results then they will help you.
Good luck anyone out there that read this far!