That's a pretty interesting idea, though I think running Time Machine on a <$200 device primarily intended for streaming video content would be tough.
Nah, the Airport Extremes were able to act as TM network destinations from their very first generation. Compared to the hardware in an AppleTV, this is very little extra processing work.
Looking at the MacBook Neo, the hardware of the current Apple TV is lower spec, the next iteration of the Apple TV will be lower spec than the Neo, but not by much, so hardware-wise, the new Apple TV could be seen as a “Mac mini Neo” that’s locked into TVOS.
No version of the AirPort Extreme could be described as being loaded with ram and processing power.
Handling TM backups would be a strain on the hardware - the heavy lifting , such as it is (really not very heavy) in TM is client-side.
An AppleTV's hardware probably would be adequate for Time Machine as a dedicated device, but the CPU and the 4 GB RAM in the current box would be a little light to support simultaneous backups and streaming. With similar hardware, a little more RAM, and an internal spinning drive, though, you easily could get a fully modernized, large capacity Time Capsule for under $500.
I just think having to buy either a Mac mini or a NAS purely for TM for one or two Mac’s is overkill, yet incremental backups are essential and Apple don’t offer a more appropriately small option (constantly having to put in an external drive to a MacBook is a bad solution - people want automation and “it does it on it’s own”.
Relying on the router provided by your Internet provider to have a USB plotting and reliably handle TM backups is very hit and miss, some do, some don’t some seem to work it crash out after the backup transfer starts.
The solution you’re talking about makes sense, but it should be cheaper.
I’ve made a load of “Time Machine” boxes for friends and family as gifts when they get a newMac (usually coincides with birthdays / Christmas ) buy a cheap Mac mini 2011 / 2012 / 2014 off the internet, put in a small SSD for the OS, a 2.5” tb in te second internal bay as the TM destination - put latest version of MacOscit can manage natively, set up file sharing for TM, turn off anything else that will eat power and performance, and get them to put it via Ethernet to their router or run it off so-fi. Slow, but TM doesn’t need to be fast.
Building this shouldn’t go over the equivalent of 200USD, so Apple should be able to sell a box that does the same for 250 -300 and still make a healthy profit.