This is a case where guide books over simplify matters to your detriment. What you are seeing on National Rail is basically correct, that there are different fare types in 1st class to 2nd/standard class.
Also it is complicated by changes in fare structures- for instance we have an upcoming 'simplification' of fares on Transport for Wales where certain cheaper fare types are to be abolished- effectively leading to a fare increase when fares are allegedly frozen.
The matter is further complicated by both Great Western and LNER (your most likely companies* for these journeys) having on the day Advance (train specific) fares.
- York to Edinburgh you can also do on Trans Pennine Express (TPE) and Cross Country. Whatever the timetables tell you about a change at Newcastle being needed on TPE that is not always correct- some trains are through services (although you get off and back on at Newcastle).
London to Bath- standard class has a variety of fare types, 1st class only has 1- Anytime Day Single £199.80 from Paddington, or £91 from Waterloo via Salisbury. However if I want to travel after 1036 today there are Advance 1st class fares from Paddington starting at £76.50
London to York - assuming LNER (Grand Central have different fares)- 1st class only has Anytime at £303.10 (any operator) or £295.40 (LNER only) and Off Peak £214 (NOT VALID FROM LONDON 0645-0758 M-F&1558-1745 M-TH).
But if I was at Kings Cross now (0718) I could get an Advance 1st (train specific) on the next train at 0733 for £199.80, a normal situation. Now I now that isn't cheap but is still £100 less than an Anytime.
York to Edinburgh in 1st there is Anytime any operator £180.90, LNER only £171.60, TPE only £138.90, or Off Peak LNER only £108,30, TPE only £102.10. But again the next train today at 0806 LNER Advance 1st £133.80, or the 0919 £80.30
But in 1st the other benefit is the lounges- so at Paddington the lounge is in Queen Victoria's private waiting room, even if the F and B offerings are mediocre.
With a pass you can use any lounge before or after travel, so if I have time and am using LNER or Thameslink/Great Northern (to say Gatwick or Cambridge) I will use the Avanti Lounge at Euston, because it is far superior to the LNER lounge at KC- especially at breakfast time or early afternoon.
However the LNER lounges at York and Edinburgh are a lot better than the KC one IMO.
Also bear in mind that full fares allow for breaks of journey and on LNER are valid for 2 days. So you could buy a London to Edinburgh LNER only Anytime 1st for £337 and legitimately have an overnight stop in York on the way- early train to York on Day 1, late train York to Edinburgh day 2. Two full days in York for essentially an extra £38 in fare.
So ultimately it's all about flexibility, and not having to think about the ifs and maybes.
Would I buy a pass for those 3 journeys, if wanting flexibility and not having to go through mental contortions?- you bet I would.