Did the ETF pump already happen?

There seems to be confusion on here about how ETFs work, and I'm also not an expert so correct me if I'm wrong but this is my understanding:

A spot bitcoin ETF would mean financial institutions would be allowed to sell shares of bitcoin they manage to customers for fees. The financial institution would hold the bitcoin and manage any wallets associated with the bitcoin. In the case of a spot ETF, the ETF directly corresponds to physical bitcoin that the institution holds.

But here's where the confusion starts:

The institutions don't have to wait to acquire bitcoin to package into ETFs, they could already have done it or be doing it. If institutions feel that bitcoin price will pump, there's no reason for them to wait until ETFs are approved to start accumulating. If the demand for spot ETFs exceed their current holdings then they would have to acquire more, but there's nothing stopping them from acquiring right now (or even months ago).

When the institution sells an ETF they're not actually selling bitcoin directly, they're selling shares of bitcoin they already own. I don't see how this transaction would affect the price of bitcoin directly. It's not new money entering the crypto space. It's new money entering the institution. The institution already bought the crypto and therefore that new money already entered the crypto space.

In other words, the institution buying bitcoin should be the driver of the pump, not the institution selling ETFs. The exception would be if ETF demand exceeds institution expectations.

Real world example:

Blackrock (maybe) has already acquired billions of dollars of crypto. These acquisitions represent demand for bitcoin and therefore drive the price of bitcoin up. Blackrock then stores these bitcoin in a cold wallet and sells shares as spot ETFs for a nominal fee. Those ETF transactions are not actually moving any crypto and therefore don't affect demand for crypto (unless blackrock can sell more spot ETFs than it currently holds in bitcoin).

Someone with more financial knowledge please help me out.