Not really much of a concern, as you would have to be wealthy enough to have spare money to put into an index fund while retaining enough liquid assets to cover your living expenses.

By making yourself the sole owner of the company, you can get around all these problems, as juridically the LLC is a separate entity from you doing all the trading. This kind of setting is legal, but practically off limits for many.

Not even remotely true. SMLLCs are disregarded entities for tax purposes, so their transactions are taxed directly to their owners. And legally, the concept of veil-piercing applies to SMLLCs generally, and especially to undercapitalized SMLLCs (meaning any LLC without sufficient independent assets to pay off all liabilities against it).