So what does it take for PBS to purge inactive members?  I have a request for a book that looks like it will time out.  When I look at their bookshelf they haven't posted any books since 2019.  Is it safe to assume they're no longer members?  If so, wouldn't PBS have done something when they didn't renew their membership?

They just end up being a la carte members.

After a certain number of requests going unanswered or timing out the member account is flagged or put on hold. PBS does not disclose the number of times it takes, but it does happen.

There are people showing as still members that I know are deceased and some that quit over 5 years ago are still showing up. One reason I think is if they are purged then membership will go down, but is that any worse than books being offered to non-members and waiting and members becoming frustrated? 

Accounts also go on hold if someone doesn't log into the site in 18 months. 

I would also appreciate inactive accounts to be dealt with better. I do think, just my opinion, all PBSers who are active, should be able to PM. 

I don't know if this is done, but it would be pretty easy to pull the people who haven't logged in for 12 months or 18 months and send them an email, and if they never answer or it bounces back then they can go as an inactive member. At the very least, pull the books from their bookshelves and if they have an automatic request on takevthat off and put their wl on vacation hold.

All good recommendations for anything over 12 months inactive I think. 
Google is getting rid of email accounts that have been inactive come December so obviously it can be done. They emailed anyone who hasn't been active recently to let them know. I then told them I had tried repeatedly more than 3 years ago to deactivate my gmail account to no avail as the system would not let me. 

As I predicted the request timed out for lack of a response.