Without this proof of work, users can be arbitrarily denied access due to overload.
With the proof of work, I think the assumption is that a legitimate user will be willing to accept a sufficient delay to make botnet DoS attack impossible.
The counter argument might be that the botnet can generate arbitrarily hard proof of work, but this isn't true. Assuming some fixed capacity by the botnet, and that the botnet needs to send some amount of requests per time in order for the attack to be effective (e.g., if they only send one proof of work request per minute, then the "bid" for the rest of the time is quite low), then there's some maximum effective "price" based on the capacity of the botnet past which a user is guaranteed access.
For example say a botnet has access to a million compute nodes, so they have 1 million proof of work seconds per second.
If there's a service which can serve 1000 requests per second, then the average proof of work seconds they have per request is 1000 seconds of proof, so as long as the user is able to provide 17 minutes of proof of work, they can get access. In reality, the user's hardware is likely more capable than the attacker's botnet (which is probably IoT devices etc.,) so the ratio is more favorable.
I find it amusing that this approach is pretty on brand for onion services, as an invisible hand type market solution from what I'd consider to be a fairly libertarian leaning group.