I have a slight quibble with the first point, though I generally agree. There are public companies out there that simply don't care about Wall St, and are run like they're private. It requires an admittedly rare combination of founder controlled/involved, no need to raise capital, strong business model, and unique culture. Think about Amazon, who have given the capital markets the biggest middle finger of all time, refusing to show a single dollar of profit because they believe the reinvestment opportunities are too great (I would strongly agree).
There's an agricultural company called Seaboard Corp that's public. Its's a $5 billion market cap company controlled by the founding family. There are no analysts covering it, they've never once held a conference call, they don't give guidance, etc. The stock is near its all-time high and has returned a staggering 22% annually since 2000. Sure, these are rare, but the market is pretty efficient about this stuff over time.
So it's possible, and I bet it would be possible for In-N-Out as well. But it would probably still be viewed as a hassle, and if no one in the family needs liquidity to buy a super-yacht, or pay a massive divorce settlement, or buy an NBA team, there's no point.