Do large banks and institutions have enough physical cash to handle multi-billion-dollar cash transactions (e.g., cash-based M&A), or is it mostly an accounting mechanism?

It is accounting bookkeeping entries.

If the banks/institutions in question have enough cash-on-hand, i.e. ledger balance large enough to do the transaction it is simply a double-entry accounting done between both institutions who have decided to partake in the trade. If there are intermediary banks involved, then same principal:

Extremely crude way of showing it:

Example: A wish to pay C, via B. A will Debit their Account Corresponding Credit will exist with B. B will Debit their Account Corresponding Credit will exist with C.