In the U.S., at least, I've never known a seller's agent to do that (having done a good number of real estate transactions). In fact, in my experience I find it to be the exact opposite. Seller's agents are incentivized to not work with buyer's agents, if the situation arises where they're approached directly by a buyer.

If the seller's agent is comfortable acting as the intermediary, and recognizing the limits of conflict of interest, then it's in their best interest to keep all of the 6% commission instead of giving half of it to the buyer's agent.

Correct, and this is something I've used to my advantage before. It's not a huge one, but when all othet things are equal...

When looking for a place in New York, we were straight up ignored by most sellers until we signed a contract with a buyer's agent. Apparently, people without an agent are perceived as window shoppers / tire kickers who are not serious about buying and whose presence wastes everyone's time. A buyer's agent signals that you have serious intentions.

Your area might be different, of course.

Interesting. Yeah, in Los Angeles, Denver, and Atlanta, it’s always worked to my advantage as the buyer. ¯\_(ツ)_/¯