I finally replaced the WiFi part of it using a 3-node Tenda Nova MW12 mesh system.
To do this, I plugged the primary Tenda node into one of the ports on the router.
Then I placed the other Tenda nodes around the house, turned off the old router's wireless function, and we were off! Well, after changing the SSID of the Tenda mesh to be the same as the old router's SSID, with same password, so I wouldn't have to update the wireless connection details on all the devices around the house...
Anyway, all was fine until I wanted to print to my wireless HP M252DW printer, from one of the computers that were still wired to the old router. My windows machine could no longer see the wireless printer on the network. Turns out, the Tenda mesh is on a different IP subnet to the old router. By default the mesh devices were on the 192.168.5.x subnet, whereas the old router subnet was 192.168.0.x.
After a lot of faffing about, it turns out that Tenda have anticipated my use case, and I just hadn't configured the mesh properly.
By default, the Tenda system is set to DHCP mode under Internet Settings > Connection Type. All I had to do was change it to Bridge mode. Then the mesh started using 192.168.0.x IP addresses assigned by the old router instead, and I was now able to print to the mesh-bound wireless printer, from the old wired network.
If anybody is interested in my thoughts of the Tenda Nova MW12 mesh, it's early days but the WiFi is markedly faster than via the old Archer C7, and the coverage around the house is now superb. Haven't had any dropouts yet. The MW12 also has a PPPoE mode, and port forwarding, so really I should be able to bin my old router completely. I will be doing this as soon as I get a cheap ethernet hub, as I need more ports than the Tenda node provides.