when the real thing that causes bureaucracy is means testing

The real thing that causes bureaucracy is the Principle-Agent problem. How do principles demand labor from agents at a cost less expensive than doing the work themselves? They need a system to efficiently monitor and evaluate the work performed by their agents prior to release of payment. So they invent a bureaucracy.

There was basically no bureaucracy with distribution of COVID vaccines in comparison.

There was absolutely a bureaucracy involved in distribution of COVID vaccines. Agents promising to collect and distribute tests and jabs in exchange for fees needed to interface with the agencies warehousing the vaccine materials and prove out that shots were going into the arms of US residents in a timely manner. There was a plethora of licensing and monitoring and auditing that occurred around the production, distribution, and payment for services rendered.

But because vaccinations have a very inelastic demand, people had a high propensity to receive the initial jabs and virtually no propensity to get subsequent jabs. This is starkly different from demand for food or housing or straight cash, wherein any individual participant will happily consume more given the opportunity and capping per-unit distribution makes a lot more sense.