Any insights on whether #![feature(let_chains)] is likely to become part of Rust or be thrown away ?
Given that the final syntax is undecided, I wouldn't recommend relying on it staying as it is today in the future. In addition, even the idea of let chains is an eRFC, something that is much easier to remove from the language than an RFC, and could change drastically before stabilization. It's more akin to rust trying something out than a feature stemming from a non-experimental RFC, which just needs ironing out before it's stabilized.
Not to say it isn't an extremely useful feature, or that I wouldn't really like it if it got into the language in one form or another. But nothing about its current status guarantees it stays constant. It will likely change at least somewhat before stabilization is even considered.