I’m pretty sure the reference to the rabbi is from the Talmud, but in any case, yes all three ideas about retribution (ECT, anni, Kath) were around in Judaism in the 1st century, so it isn’t surprising that the fire is described that way.
As to Lightfoot’s reply to 9:49-50, I notice that so far as you quoted him, he says nothing about salt being ideal (or the best of things), and that having salt in our heart leads us to being at peace with one another. But Jesus says those things when talking about everyone being salted with the unquenchable fire of Gehenna (which Lightfoot does acknowledge is the fire being referenced).
Jesus not only says nothing about too much salt being bad, but references an apparently common saying: if salt becomes unsalty, with what will you season it? (In a related saying elsewhere He adds that unsalty salt is regarded as worthless and thrown out to be trampled underfoot by men.)
So, just as might be expected from the idea that salting is the best of things, Jesus doesn’t talk about having too much salt, but about not having enough.
I’m not going to dispute if Lightfoot found some rabbis talking about something being “fired” as a way of describing being ruined by too much salt (although I’m curious about the contexts if so); but if Jesus was referring to that idea at all, He was refuting it (and by means of another one of His classic unexpected reversal methods common to the Synoptic reports).
This applies just as much to the disputed addition to 9:49, by the way, where every sacrifice is sacrificed with salt: the sacrifice is washed (in a basin which typologically resembles a sea/bath of fire), salted, and then cooked, as a three-fold way of offering it cleanly to God’s acceptance (and eating by His servants). I can’t tell from your quote if Lightfoot mentions this very common textual variation, but just in case I thought I ought to anticipate a potential objection (since sometimes people will cite that textual variation as if its possibility of being original somehow automatically changes the meaning to a hopeless punishment situation).