You say all of this is just your guess. But you don’t need to guess, and your guesses are wrong. The reason they struggled to raise collateral is all bids for gilts disappeared in the immediate panic over the Governments budget.
You can’t post cash if you can’t sell your gilts.
The BoE stepped in as a buyer in the short term to restore the market to their usual function…
Again, don’t confuse liquidity and solvency. Cash and assets are not the same thing and turning assets into cash is usually easy but not when the market is dysfunctional, like when they’re spooked by a UK government with reckless fiscal plans.