How exactly is wait-listing processed, in general? Why does the T&f page over the Library Platform does not display the wait-listing banner; as visible over few of the other resources?

Collection level waitlisting is yet to be integrated into the platform (see: T189592), right now waitlisting works only on the partner level. As for Taylor & Francis not being waitlisted, it has two other collections with accounts available.

I assume that when a resource-provider enters into a contract with Wikipedia; the number of folks who can access the request at any given amount of time is pre-defined. If such is the case, why is not a running count kept? For example:-- Quota--200; Have access--186; Applied but yet to be accepted by coordinator (Waitlist)--24.

Number of accounts available for distribution for a partner (Find 'Accounts availability' or 'Accounts available') or for collections are displayed on every partner page under 'Metrics'.