Why do electric heaters say never to use an extension cord, when a properly sized cord would be fine?

The two non-electrical reasons matter as much and get less attention.

Tripping. A heater is a heavy hot object on the floor and the cord runs across a room. Pulling one over is a fire and a burn, and it is a far more likely event than an overloaded conductor.

Placement. The reason people reach for an extension cord is that they want the heater somewhere without an outlet — often exactly the places you should not put one. Near furnishings, in a corner, against a bed. The cord enables the bad position, and the bad position is the actual hazard.

That is why the advice is absolute rather than conditional. A conditional warning would be technically better and would be followed by people who already understand the conditions, which is not the population the label is written for.

You have essentially answered it: a correctly sized cord in good condition really would be electrically fine. The warning is written for the general case, and the general case is bad for reasons that go beyond the conductor size.

What makes heaters specifically dangerous with cords:

• They draw near the maximum, continuously. Most appliances are intermittent or light. A heater pulls close to the circuit's rating for hours without pause. That is the worst possible duty for a marginal connection, because heat accumulates rather than dissipating between uses.
• The failure point is usually the connection, not the wire. A slightly loose plug in a socket, or a worn contact in a cord end, has resistance. Resistance times a large continuous current is heat, right at a plastic connector. This is how these fires start, and a heavy cord does not protect against it.
• Cords get covered. Under a rug, behind furniture, coiled up. A coiled cord carrying a heavy continuous load cannot shed heat and gets substantially hotter than the same cord laid out.

The cord in the wall does not have plug connections, is not coiled, and is not walked on.

If you genuinely cannot avoid one, the version that is least bad:

• Heavy gauge, short, and rated above the heater's draw with margin.
• Fully uncoiled, laid out where nothing covers it.
• Nothing else on the same circuit, and check what else is on that breaker.
• Feel the plug and the cord after twenty minutes. Warm is a warning; hot means stop.

But the better answer is usually one of: use a heater on a circuit that reaches, have an outlet installed where you need it, or use a different heating method for that spot. An added outlet is a small job and it removes the problem permanently.

Worth noting the same reasoning applies to power strips, and people are even more relaxed about those.

A strip adds two more connections and is often the cheapest available. Manufacturers of heaters, air conditioners and similar high continuous loads all say the same thing about them for the same reason.

If a strip already runs to where you want the heater, that is a sign the room is short of outlets rather than a solution to it.