You can theoretically create a Thunderbolt 1 or 2 to USB-A adapter … but I don't remember there ever being any other than USB-A ports built into Thunderbolt 1 and 2 docks.
Other World Computing sells a "Thunderbolt 2 Dock Solution", which as far as I can tell, simply bundles Apple's bidirectional Thunderbolt 3-to-2 adapter and one of OWC's Thunderbolt 3 docks. That dock provides 5 USB-A ports and 1 USB-C (USB protocol only) port, among other things.
Other World Computing – OWC Thunderbolt 2 Dock Solution
It seems possible that you could use the same approach to connect other docks that (a) are Thunderbolt docks, which (b) have their own power supplies. (The Thunderbolt 3-to-2 adapter does not convert other protocols or carry any power from one side to the other.)
Such a setup would be bottlenecked by the speed of your Thunderbolt 2 port and by the display support of your MacBook Air (which can support a UHD 4K display, but not a 5K/6K one).
The question would then be how much money you want to sink into keeping an old, partially-broken MacBook Air running – though, presumably, once you did upgrade to an Apple Silicon Mac, you could reuse the Thunderbolt 3 dock with it.