Primary dealers have a privileged position, so there is value in being registered as a primary dealer. - https://economics.stackexchange.com/a/13450/8387
Who Can Participate?
All auctions are open to the public. The following Treasury services enable various categories of investors to participate directly:
TreasuryDirect accounts: Individuals and various types of entities including trusts, estates, corporations, partnerships, etc. ...
TAAPS: Institutional Investors
Additionally, according to the New York Fed, being a primary dealer isn't a public endorsement.
In light of the foregoing, third parties are reminded that the designation of an entity as a primary dealer by the New York Fed in no way constitutes a public endorsement of that entity by the New York Fed, nor should such designation be viewed as a replacement for prudent counterparty risk management and due diligence.
It looks like primary dealers get their voice represented on surveys that may affect monetary policy.
[primary dealers] provide the New York Fed’s trading desk with market commentary and market information and analysis helpful in the formulation and implementation of monetary policy
So do primary dealers apply for this reason? Or is there some other reason in applying to become a primary dealer?