Try power cycling the modem (I suppose you might have done this already!)
Yes, I have done this many times.
Try booting the system in safe mode with networking and check if the issue can be reproduced
I should have mentioned this earlier. I did try this and the issue persists.
Check for any firmware updates for your modem
No firmware updates for this model. This model is fairly old. Like 5 years old!
Incase you haven’t tested other devices, do that. If the DNS fails on other devices too, then it’s more likely an issue with your router or ISP.
I will update on this when I get hold of a friend’s laptop.