No (at least, not in general)

Pathfinder 2nd Edition is very explicit about using traits to define whether something is or is not another thing. Anything which refers to an "Unarmed Attack" is referring to an attack (i.e. something with the Attack trait) which has the Unarmed trait. Attacks being natural does not include any extra traits (in fact, Natural isn't even a game term, and its inclusion could be completely removed with no effect).

Of note though, many of these spells are on-par or close to for the level of spell they are. Additionally, they include the ability to heighten them to higher levels, which appears to normally vastly increase the damage dice of them.

The spells also set your statistics, so even a Sorcerer with 10 Strength and 10 Dexterity could have a much higher AC/damage bonus than their stats would normally allow, which makes these polymorph spells useful in many cases.

Taking the Dinosaur Form example, it can be heightened to just 1 level higher, doubling the damage - this gives a 10th level caster the ability to give themself a +16 attack which deals 4d8+6 damage; which is roughly equivalent to a +4 longsword wielded by a similar-leveled Fighter with 18 Strength.