Not needing a visa and not being able to stay for as long as you wanted are two different things, what you're looking for is "free movement". If you hover "ease of travel" on the best second passport page, they show this with "freedom of movement > visa free > visa on the border".
Eg for a lot of asian countries, we european can go at will, without a visa, as long as you stay for 30 days or less. It's not a visa on arrival either, they merely stamp your passport with the entry date. But if you want to stay longer, you need to ask for a visa (and for some countries, you can do that visa on arrival, in the airport).
However for the US, us european still need to apply to the ESTA, which is a visa in anything but in name. I'm not sure why the EU is not forcing the US to drop it for us, or force it back on US citizen coming here, because this seems to be a trickery against the "no visa required" deal.
Is this really true?
I had the impression I can't simply go to USA whenever and as long as I wanted, which sounds like needing a visa to me.