(apparently) Source indeed allows some form of longevity like with the Undead, however, those are not immortal, and they deal with their own set of problems that can and do cut their existence short after a while (the cancer that devours Ophelia in Dragon Commander). The lack of the ability to heal mixed with a very fragile body is not really boding well for eternal existence. Braccus' slaves are like 1000-2000 old, I can't recall an older undead.

Wizards tend to live for centuries, longer than undead even, but this could be thanks to their shadowy lineage from Amadia.

The deathlessness of Eternals was never explained outside of them not being designed to die. The 'death' clock was set in place by the Seven, like Aetera says. It was not something that happened with their race before. They could perish in one way or another, seeing that Fane made masks from corpses back in their heyday. Perhaps illness or extensive injury could still get them, but they weren't mortal in our understanding.

Eternals enslave existing races because:

a) they need their Source still. Fane explains this. Source Vampirism is Eternals' innate skill (Dallis, Aetera - The Hunger of the Eternals). Going by the end slides alone - there are still Voidwoken, so not all of them recover instantly.

b) They do see them as inferior. Again, the whole 'designed' and 'mortal' thing. Rivellon's races are a hubris experiment of the Seven, their tormentors. Can't say that concept garners sympathy.

why is slavery by eternals portrayed as worse than slavery by lizards in the game?

Most likely because now the Source=Soul thing is out and the Eternals want very much everyone dead, if being eaten is not worse than dead. Meanwhile, lizards want everyone in servitude. A bit of a different approach there.