Routers are already heavily overloaded with wireless and routing duties.. and in the more recent incarnations needing to handle mesh for example.. while all the time maintaining perfect connections to networks from all PC and device manufacturers.
Plugging a USB 3 hard disk into the back of one of them and expecting it to work well for Time Machine is adding to the range of accessories.. when it has to cater not just for Mac but windows and linux and all the mobile devices.
The synology router does very well at the storage side.. since the OS was really designed for that.. but I much prefer the Asus as a router. All in one products compromises some aspect or other. That is why in the pro world you keep all these functions to separate products.
So probably it sounds like I should consider putting in a good wifi router, and a separate NAS for the Time Machine software to work on.
Ideally yes.. the linksys routers you have might work fine for you.. since they cannot do Time Machine as well then adding a NAS would be a better solution .. perhaps.
Cost could be a factor. A USB disk plugged directly into the computer.. as Kappy alluded to.. is faster, cheaper, and much more reliable cf network backups. For a desktop Mac I would certainly recommend this as well.
But for a laptop it is really neat to automate the backup over wireless on home network.. so it is done without the hassle of plugging in a drive every time.
So plugging the same USB drive into the router and sharing it with the network sounds great. The reality is without Mac OS and using linux mostly routers do not always behave as reliably.. and backup over network is much more problematic than it was under Snow Leopard or Lion where it worked very well.
So I would rate..
Local Disk .. best by far.
Synology NAS or Time Capsule. Best of the network solutions. Lets say 50% of local disk reliability at best.
Other brand routers with USB.. I would put synology router top.. I will give that a 45% of local disk.
Asus 20%
Most other brands or even cheap end NAS would rate between 0-20% depending on how well they support TM and how reliably.
I have used DLink products with great success. I have successfully used NetGear, TP-Link, Arris, and Motorola routers at one time or another with my several Macs, iPhones, and iPads.
From Kappy's list the vast majority do not and never have supported Time Machine.
(I also despise D-Link having very poor experiences with them.. but each person sees the world in somewhat different ways)
He is talking about them as routers.
Some Netgear do support Time Machine.. it is horribly slow..
Only the very latest routers with 1.4 or 1.8ghz processors are really up to the task of doing both routing and NAS in the one box.. and most of them don't have sufficient RAM to make it fast as a real NAS. Synology and Asus will do about the same speed as the Time Capsule.. say 50MB/s which is adequate. There are a couple of exceptions. Linksys WRT1200AC and up are amazing at USB speeds on a network.. exceeding 100MB/s which is close to saturation for gigabit. But Linksys did not bother with Time Machine... which makes them useless to the Mac user.
I had someone else also recommend the Synology DS 211.
Any of the synology series.. they all run near identical management software.. DSM.. but the underlying firmware is of course based on the processor type. 211 is old.. 2 means two disk.. 11 is year.. 2011. You can get 218 now. higher end processors get higher numbers. but any of the ones introduced in the last couple of years should be able to saturate gigabit.. i.e. read at 100MByte/s .. write speed is probably a bit less as it is more intensive task.
Prices go up dramatically with the better models though.. DS918+ for example.. is $550 US on Amazon. Disks are on top of that. So USB might be a good compromise with a Synology router.
Also a better backup software like Carbon Copy Cloner will be much more reliable.
If you happen to have a desktop Mac you can also use it for Time Machine backups with a big external drive.
https://www.howtogeek.com/330288/how-to-set-up-your-mac-to-act-as-a-networked-time-machine-drive/