A year ago, Elon Musk’s Tesla Inc. was thriving, the unrivaled leader in electric cars with a valuation that had peaked above $1.2 trillion.

Today, it faces problems that have gashed Mr. Musk’s wealth and shaken faith in the company at the center of his business empire. Rather than looking like one of the world’s fast-growing tech giants, which is how it was valued by investors, it now looks more like a car company.