So the question is, which provider will notify you that the advance fares are available to purchase, not necessarily that they are available to purchase from them?
Every system seems to have flaws. The Trainline one effectively screenscrapes itself to detect drops in fares returned from searches. It then assumes that this is the release of Advances.
The LNER one doesn’t reliably email people (see Twitter for complaints), but then when it does, the fares often aren’t available on the LNER website, which seems to lag The Trainline, other Trainline powered and WebTis sites by anything up to about half an hour.
By the time they do appear on the LNER site, the often very low quotas might have been exhausted by purchasers from other sites.
The LNER one will also sometimes refuse to allow an alert to be set up, saying that tickets are already on sale, when they’re not - presumably a function of situations where the Advance On Sale page doesn’t correctly show that (say) weekend dates within an otherwise on sale time frame are not in fact available.
Again, see Twitter for complaints.