If you're compute-bound, then Python async (which uses cooperative scheduling similar to green threads) isn't going to help you. You get parallelism, but not concurrency from this progamming model; only one logical thread of execution is running on the CPU at a time (per-process), so this can only slow you down if you are CPU-constrained.
The standard usecase of a sync API backed by a local DB with low request latency is typically going to be compute-bound.
This is covered in the Django async docs (https://docs.djangoproject.com/en/3.1/topics/async/) and also in green threading libraries like gevent (http://www.gevent.org/intro.html#cooperative-multitasking).
The case where async workers are interesting is for I/O-bound workloads. Say you're building an API gateway, or your monolithic API starts to need to call out to other API services, particularly external ones like Google Maps API. In this case, the worst-case result is that the proxied HTTP request times out; this could block your Django API's work thread for many seconds.
In the async / green-threaded model, this case is fine; you have a green thread/async function call per request, and if that gthread is blocked on an upstream I/O operation, the event loop will just start working on a different API call until the OS gives a response on the network socket.
Essentially, there's no reason to use Django async if you're doing a traditional monolithic DB-backed application. It's going to give you benefits in usecases where the standard sync model struggles.
(Note, there's an argument that you might want green threads even in a normal monolith, to guard against cases like "developer accidentally wrote a chunky DB query that takes 60 seconds to run for some inputs", but most DB engines don't support one-DB-connection-per-HTTP-connection. There was a bunch of discussion on this topic a few years ago, with the SQLAlchemy author arguing that async is not useful for DB connections: https://techspot.zzzeek.org/2015/02/15/asynchronous-python-a... although asyncio support was added: https://docs.sqlalchemy.org/en/14/orm/extensions/asyncio.htm...)