DNS problems with ASUS router

I have an ASUS RT-AC55U router purchased within the last 2 years that is experiencing continually more confounding problems.

Situation

More and more frequently, the following conditions suddenly present themselves:

My Windows computers, despite claiming an internet connection, are unable to connect to any web pages via 2.4G, 5G, or wired connections. Chrome gives the error DNS_PROBE_FINISHED_BAD_CONFIG.
On my Android devices, both the 2.4G and 5G connections become extremely slow. If I disconnect from either wireless network, I am unable to reconnect.
I am unable to access the router's internal settings page via wireless or wired connections, or access any other device on my local network.
If this happens while my work computer is connected to the wireless or wired networks and tunneled into the company's network via VPN, everything functions normally on my work computer. That is, I have no problems accessing web pages or the company's internal pages.

What I have tried

Unplugging and replugging the router (duh). This usually fixes the problem for a while.
Factory-resetting the router and rebuilding all settings from scratch. The only things I changed from the default were the network name and password and the router's login credentials. This reset didn't seem to have an effect on the timing or nature of the issue.
Making sure the router's firmware is up to date.
ASUS support. They told me to do the previous three things (duh). I'd prefer not to pursue their avenues further.
Unplugging the HDD from the router's USB 3.0 port. This doesn't seem to have an appreciable effect on the timing or severity of the issue.
Disconnecting all wireless and wired devices from the network then slowly adding them back one by one. No particular device seems to trigger the problem.

I am hard-pressed to think of anything that changed with my home network that specifically correlates with the onset of this problem. Any tips or tricks are appreciated, and I'm happy to provide more information if necessary.

Additional info

In response to comments, I've tried reconfiguring the DNS settings on the router and am now working on keeping it cooler. It's working right now, but I'm not holding my breath. It even stopped broadcasting the wifi signal entirely today, so I'm now fairly sure it's a hardware problem. I'll keep this up in case others find it useful, and I'll update it with changes to avoid something like this.

Update

ASUS warranty service sent me a refurbished RT-AC55U after diagnosing the problem. Can't complain about any aspect of the process except having to pay to ship the router to the repair center.

Your router is failing (physically)

I've seen the scenario you've described many times, particularly with consumer-grade networking gear. One moment it works, the next it's exhibiting unwanted behavior. Usually a reboot fixes things for a while, but eventually the reboots must happen too frequently to remain a viable solution.

Even solid-state technology can wear out (or worse, be of low-quality and fail prematurely). Someone else could provide detail on exactly what components in these devices are likely to fail and why, e.g. capacitors leaking fluid, but the fact stands that it does happen.

If you have access to another router, try temporarily putting it in your network. Even if not, if I were in your shoes I'd get a new unit (or make a warranty claim on the existing one).