If you just want to copy files to the disk, that should be possible.
You might need to adjust the read and write permissions on the Linksys to allow the Mac access. It sounds like you currently have read permissions but no write.
I have now tried with a few Linksys routers and find them difficult to use with the Mac.. especially getting the Mac to actually be able to write to the USB disk.
There is a big difference however between copying files to a disk and being able to get Time Machine working.
On Mojave you do not need to use AFP .. you can also do Time Machine over SMB protocol.. but Apple did not make it quite that simple. Any router that is going to work with Apple version SMB will have to have a few special tweaks.. since Apple wrote their own version of SMB.. which is predominately the Windows network protocol.
You can either replace the Linksys.. with a router that is Time Machine aware.. Bob listed the two major ones.. Asus and Synology but you can also get some Netgear.
However I urge caution as my testing with these alternative routers has certainly given more issues than it should. I would trust the Synology as better than Asus at this point.. although I prefer the Asus as a router.
You can add a NAS to the network.. these are generally expensive.. but Synology NAS have a good reputation with Time Machine which is why I would tend to trust their router as well.
As a really cheap alternative.. a second hand gen4 TC with a replacement hard disk could be made to last several years beyond its normal life.. In the area I live you can buy these on ebay for around $50. You plug it into the main router of whatever brand and use it only for Time Machine.. it works very well. I have one configured with a 5TB drive.. and you can easily use any low power drive.. up to 8TB.
