Is it seriously impossible to add a Time Capsule to an existing network via wifi?

Usual setup nowadays with an existing Wi-Fi would be a NAS device with Time Machine support, such as a Synology DS220+ or ilk. This as Apple exited the Wi-Fi market a while back, as you well know.

If you want to run Time Capsule wired, and the existing Wi-Fi coverage is good, turn off the Wi-Fi on the Time Capsule.

If you want to run multiple active Wi-Fi devices, you’ll likely either need to configure those as access points (APs), or as separate networks and subnets. I’ve found AP configurations usually work better, as roaming from one Wi-Fi to another is far less of a hassle with a multiple APs than with multiple Wi-Fi routers.

Configuring Time Capsules as APs can work, but it also means having DHCP and other services available elsewhere on the local network.

I had Wi-Fi coverage and reliability issues with the older Time Capsules I was working with, and have been migrating off those, using APs from another vendor (Ubiquiti), and running the remaining Time Capsules with Wi-Fi disabled.

Is it seriously impossible to add a Time Capsule to an existing network via WiFi?

Yes and no.

Yes, if you are trying to add the Time Capsule to a network provided by a 3rd party router.

No, if you are trying to add the Time Capsule to network provided by another Apple router.

Is it seriously the case that Time Capsule, a wifi device, is only capable of joining an existing network via ethernet?

Yes, if you are trying to connect the Time Capsule to a network provided by a 3rd party router.

No, if you are trying to add the Time Capsule to network provided by another Apple router.

Apparently other manufacturer's routers can accept a hard drive and support Time Machine, but despite my searching, I can't find any documentation on which routers will support that.

Synology and the upper end Asus routers. There may be more, but no one has reported what those might be.

https://www.synology.com/en-us/products/RT2600ac https://www.asus.com/support/FAQ/1011283/

All I wanted was to be able to do wireless backups.

Connect the Time Capsule to your WiFi router using Ethernet and turn off the WiFi function on the Time Capsule, so you can back up over your existing WiFi network.

Even if you could connect the Time Capsule using only a wireless connection......(you can't on a 3rd party network), this is a horrible way to try to back up, since you will be dealing with a double-hop wireless situation. Backups would take twice as long this way and there will be double the chances of an error occurring in the process.

Things aren't as bad when the Time Capsule connects to another Apple WiFi network, since devices can back up directly to the Time Capsule over the WiFi and avoid a double hop.

Keep in mind that the Time Capsule was designed in 2012, and that Apple made no hardware improvements to the product up until the time it was discontinued over 2 years ago.

Backups to the Time Capsule, slow they might be, might be OK, but no one in their right mind would set up the Time Capsule as their only backup plan.

The simplest, cheapest, fastest and most reliable way to back up is simply connect a USB drive directly to your Mac.

Pricing... Newer equipment can be more expensive than a ~2013 era Time Capsule yes, but then the (loaded) ~USD$1000 Synology DS220+ I’m presently prototyping is 24 TB with 2 x 12 TB high-spec HDDs and with support for RAID-1 or JBOD, versus the 3 TB single-spindle HDD Time Capsule at ~USD$400 (~2013 price). And the Synology has add-ons that can do a whole lot more than can or does the Time Capsule. Many (most?) of which are free.

Not recommending this......because I have not tried this....... but WD MyCloud claims to be compatible with Time Machine.

https://shop.westerndigital.com/products/cloud-storage/wd-my-cloud-home#WDBVXC0020HWT-NESN

Build your own, then?

https://www.freenas.org/about/features/

Not that I’ve tried this, but FreeNAS does indicate Time Machine support.