Setting up a ticket alert on LNER is definitely the correct thing to do - the way train tickets work, there are 2 different types of ticket:
advance: these are tickets for one specific train, there are a certain number per train at each price level and once one price level sells out, the next price level will become available for booking.
off-peak/anytime: these are tickets that can be used on any train on a certain day (sometimes with time restrictions eg any train after 09:30) and the price level is set whether you book 3 months or 3 minutes before the train leaves
by setting up a ticket alert, you will be notified when the cheapest advance fares go on sale - the faster you then book, the better fare you will get (they very very rarely go down in price)
On this route, the advance tickets are sold by LNER - scotrail do not offer advances between edinburgh and glasgow. Therefore, LNER is the only ticket alert service you need to sign up for.
You are correct that like airline tickets, it can be cheaper to buy separate tickets for the same journey - this is known as split ticketing, and there are plenty of explanations for how this works in more detail, but essentially, you could buy a Newcastle to Edinburgh ticket and an Edinburgh to Glasgow ticket, and you can travel exactly the same as though you have just a Newcastle to Glasgow ticket, with the only caveat being that your train must stop at Edinburgh (but there is no requirement to get off and back on the train etc)
Unlike air tickets, however, every rail website offers (almost) every ticket sold for railways, so there would be no need to use the scotrail website since the lner website is fine. However, what you can do is use a website such as
https://tickets.railforums.co.uk/ which will automatically show you any split tickets that will be cheaper for your journey than a through ticket, as they can sell every ticket the same as the LNER website
However, if you buy these tickets as soon as they are released, it is very unlikely that split tickets will be cheaper, so your best bet is to wait until LNER email saying the tickets are released and then book whatever services you want - as these will be advance tickets, you will need to take these exact trains when you do the journey so ensure that you know what times you want to travel at, as changing the times will cost money.