For most liquid toiletries, it's actually difficult to buy "travel size", i.e. less that 4 fl oz. But for toothpaste, this is a standard size that you normally buy, so it doesn't add much of a burden.

Yes, it's easy. You just buy a travel-sized tube at pretty much any chain drugstore in the US. (Or, as is the case with many people, use the trial sizes your dentist gives you when you're in for a cleaning.)

I thought toothpaste was fine because it's a paste, not a liquid, gel or aerosol.

Unless of course you buy the "gel" type.

"You are allowed to bring a quart-sized bag of liquids, aerosols, gels, creams and pastes in your carry-on bag and through the checkpoint. "

YMMV of course as to how strenuously this is enforced in practice. My observation is that in Europe (at least in the UK) the separate quart baggie taken out of luggage is actually more routinely and more strictly enforced than in the US. And I somewhat doubt that a squeezed down tube of toothpaste in that baggie is typically going to cause an issue. (But in the US, I use travel-sized toothpaste and have TSA Pre so I don't have a lot of recent data.)