So what does a buyer's agent actually do between the showing and the closing table?

The sequence, roughly, is search, then evaluate, then negotiate, then shepherd the contract to close, and the last one eats the most hours.

Search means pulling listings, yes, but also calling listing agents about places that aren't a clean fit yet, and telling a buyer that three of their five must-haves don't exist at their price. Evaluate means walking a property and noticing the roof age, the sloped floor, the septic in a market on wells. Negotiate is price, but it's also who pays for the repair the inspector found, and how many days the buyer gets for financing.

Then the contract goes live and the calendar starts running. Inspection window, appraisal, loan commitment, title work, walkthrough. Each of those has a date, and a missed date can cost a buyer their earnest money depending on how the contract is written in that state. A lot of the job is chasing a lender for a document and a title company for a payoff figure so nobody blows a deadline.

One piece you didn't ask about: the representation agreement does describe the services and the fee, and it also usually has a term length and an exclusivity clause, which is the part that surprises new buyers more than the fee does. If you go get licensed, read your own brokerage's version of that form early. You'll be explaining it out loud to nervous people at kitchen tables, and being fluent in it is most of the trust you'll get in the first year.