In my opinion, there should only be two types of tickets available for all flows - off peak (anytime after morning peak) and anytime (self explanatory), which would infinitely simplify ticketing on the railway. Off peak fares should "meet in the middle" between the current advance and off peak prices, which would mean the large majority of passengers would see a fare reduction, which would induce more train journeys.
How many of the long-distance operators could cope with inducing any significant number more journeys? The "fares trial" scam that LNER have implemented has been because they couldn't cope with the number of off-peak passengers and so needed to squeeze the numbers down.
GTRs revenue is roughly equivalent to LNER+AWC+1/2GWR so most people would discount Intercity revenue as not significant.
I don't know where you've got your stats from, but ORR7223 gives income in 2023/24 as:
Total: £10,786m / GTR £1,530m / LNER £861m / AWC £1,088m / XC £471m / GWR £1,129m / GA £687m
That puts LNER and AWC comfortably more than GTR. If we estimate 75% of XC and 50% of GWR and GA income is for intercity services, that gives a total revenue for intercity of £3,210m, or about a third of the total for all franchised operators, which is very definitely significant.
On the day advances are all very well, but there is no guarantee there will be one available at an affordable price when one needs to travel.
On-the-day advances are purely a gimmick to allow the TOC to avoid sharing revenue with other operators, and have no place on the railways going forwards.
And also to remove the embarrassment of the annual fares round - you can surreptitiously tweak with quotas to increase fares without it being visible if everything is Advances. Nobody ever talks of the Ryanair annual fares round.
Quite. We want fares to be transparent and to be able to look at price increases over time and report them fairly. Advance purchase tickets are like a magic black box where we can't see what the allocations are for each fare, so it allows the railways to impose stealth increases above and beyond any increase to the sticker price by reducing the quotas of cheap fares.
Advances are also a very simple ticket type: booked train only, if you miss it then you've paid £100 for a useless piece of orange cardboard.
Except that if you are making a connection, the connecting service may or may not be covered by the same rules about booked train only.
And if you miss a booked train because of a delay in an earlier leg of your journey then your ticket will still be honoured.
So even there, the rules aren't quite as clear cut as you suggest!