how they want to make those attesters trustworthy

It’s all derived from hardware security modules like TPM. It’s not impossible to exfiltrate private keys from these devices, but it’s difficult and expensive, involving de-lidding the chip and carefully reading electric charge values from individual flash gates. Not out of reach for a sophisticated state-sponsored targeted attack, but certainly puts Evil Maid and other opportunistic attacks out of reach.

As for how original integrity is established, that’s done by saving the public key or equivalent while the device is in the possession of the trusted entity.