The problem is that his payout, like the rest of his fortune, is in exceedingly overvalued Tesla stock. So using that to finance Twitter means selling, and every time he sells the price takes a hard dip because Tesla investors know they're standing on a soap bubble and they are extremely nervous about it bursting. Any sudden uptick in sales pressure is liable to cause a small avalanche of investors abandoning ship.
The process of buying Twitter alone cut his net worth by half because of how much it cratered the Tesla stock price.
Imagine being the wealthiest person on the planet and deciding to burn half of it to ensure that Nazis have an audience for their hot takes. Elon could have bought entire parks of grass to touch instead.
It wouldn't kill Tesla, per se. A company's stock price only really matters insofar as it helps them to carry debt. The company doesn't actually directly gain or lose money based on the stock price. What it affects, primarily, is the shareholders of the company.