The point is, who PAYS the agent? It's not (in almost all cases in the
US) the buyer. It's the seller who pays. The lower the price, the less
commission for the agent. That's a fact of the real-estate game.

I agree with half of this. The commission is a function of the sale
price, so the buyer's agent typically has an incentive to push the
price higher.
However, who pays the agent is irrelevant, except with regard to the
separate issue of how much cash you can bring to the closing. The
agent's incentives are dictated by what actions they can take to
increase their income, not by who writes them their check at the end of
the day.

Brian rightly points out that buyer's agents have an incentive that
pushes the other way: to treat you right so they get referrals.
OTOH, in a large hot market, agents will rely on referrals to differing
degrees. Some agents will be able to simply cater to ill-informed
buyers and treat them poorly, not caring much about referrals. Agents
with large cheesy ads in the paper are prime suspects here.