Why doesn’t the stock price fall after the record date for a company splitting stock?

That's not how splits (or dividends) work. If you own one share of stock worth $100 and it does a 4-1 split, you will (eventually) have 4 shares worth $25 each. Anyone who buys stock before the split is effective gets the $100 shares; anyone who buys shares after the split is effective gets $25 shares. There's no arbitrage opportunity.

Same for a dividend. If your $100 stock pays a $5 dividend, after you the dividend is effective you will have stock that's worth $95 and (eventually) $5 in cash. Again, there's no arbitrage opportunity. Either you get $100 stock and a dividend or $95 and no dividend. (There are timing differences between when the dividend is announced, effective, and paid, but that's not germane to the arbitrage question).

Stock dividends have the same dilutive effect on the original shares, so a "3 share" dividend is the same as a 4-1 split from the shareholder's perspective. The original share represents a fourth of the ownership that it did, so they are worth 1/4 of their original value, with the other 3/4 "given" back to you in the form of additional shares.

The difference between a stock dividend and a stock split is that with a stock dividend, the _company pays cash for the new shares. So instead of just splitting the ownership into smaller pieces, the company uses cash to provide additional pieces to its owners. One benefit of this is that owners will benefit from future growth, versus a cash dividend where the cash is completely out the door.

As in most cases of splits, the stock price history is adjusted to the new stock volume and this avoids dramatic movements in the stock price.

Selling shares between the record date and the distribution date also sells the right to claim the special dividend that was afforded to the owner on the date of record. If you buy shares during this period you also participate in the split.