It used to be that this served to identify passengers who had "SSSS" (selectee screening) in case they were missed by the Travel Document Checker. Since the SSSS has (thankfully) been greatly reduced, the need for the double check seems highly questionable.
There are other problems:
1. With the widespread deployment of millimeter wave scanners, airports can't decide if you're allowed to have your boarding card in your hands. Some allow you to hold it in the air; others insist that it be x-rayed (in case you had concealed a knife inside your boarding card.)
2. Electronic/smartphone boarding passes are much more common, and also throw a monkey wrench into this system.
3. The TSA inspection of boarding cards at the gate is absurd. Since the overwhelming majority of US airports have bar code readers, and I don't know of any without computerized boarding, the verification that a passenger has a valid ticket has already occurred.
At Atlanta, it was explained to me that the gate boarding pass check was to ensure that the appropriate TSA "squiggle" or highlighter had been placed on the boarding pass, indicating that a travel document inspector had reviewed it. But mine had no "doodling" since it was issued at an overseas airport. I had to show her how the issuing location was printed on the boarding pass. The "doodle" varies across airports, and Atlanta has a majority of connecting passengers, so it's particularly ridiculous. It's busy work.
On a San Francisco-Frankfurt flight, we United inspecting boarding cards and passports, then the TSA, then Customs and Border Protection. I felt like I was in the third world! Boarding that 747-400 took a really long time!
Many European airports have bar code readers at the entrance to the x-ray/metal detector. This is in lieu of the "Travel Document Inspection." They don't do a passport verification; just the boarding pass.
Canada always seems to have the greatest number of boarding pass/scanning stops. Perhaps that's why it can take forever to get through US Preclearance in Toronto. This applies even if you have NEXUS and are flying in first/business class. It's a hassle.
I think that smartphone boarding passes are the key to getting through quickly. The TSA doesn't seem to know how to read them (except scanning them), and one can easily fumble and shut off the phone. By the time you find your phone, start to turn it back on, etc., the agent has given up and decided it's not worth his or her time.