There is no additional cost. The gate agent is already checking boarding passes. You would simply present your photo id when presenting your boarding pass.
Now if you required id to access the secure area, then yes, someone has to do that, but then less people in the secure area.
The security delays in the USA are not from the id checks, but from the security screening itself.
I don't see the problem at all. Why shouldn't you have to show id to show that it is your boarding pass..??
There will be a cost. You need to make the process robust enough to spot fake ID's, because the problem people you are targetting by ID checks are those with both the means and incentive to spend money on acquiring a higher quality fake IDs. Surely for criminals and terrorists this would be step 1? In the US trained TSA agents check the ID and use (is it UV?) lights to examine passports (not sure about licenses). I assume the agents and equipment are good enough to spot fakes. But this certainly has a cost to it. Your average FA who is trying to get a flight away on time is not going to be as thorough.
In my recent experience in the US sometimes the hold up has been with the ID checking/signing BP and have breezed through security screening. In fact at one airport last year I was departing from there was a 20 person queue for ID checkpoint and 2 people in screening! (of course if the ID checks weren't as thorough, those people would be waiting at security screening instead!)