Bus tickets - online advance purchase vs on-site day of travel

The Leuchars-St. Andrews bus is a feeder to the Leuchars train station and sort of equivalent to a city bus. I used it twice in the summer of 2019, and it didn't see particularly busy. Nor did the two East Neuk buses I took, one of which was probably the X60. Although I can't say it's absolutely impossible for one of those buses to be full, it seem unlikely under normal circumstances. Others may be able to provide a more reliable answer.

There are some long-distance buses in Scotland that run somewhat infrequently. I've heard of some sell-outs and wouldn't wait until the day of travel to buy a tickets on those.

You can always do test transactions, stopping before entering payment information, to see whether buses you're interested in are sold out for today, tomorrow or the next day. If you aren't required to specify a travel date and time, you can assume you aren't buying a guaranteed seat on a specific bus when you buy a ticket. I don't know whether you might run into that on the routes you're planning to take.

A North East Fife Day Rider is valid from Monifeith (just north of Dundee) to Glenrothes and across to Newburgh, so on all your buses.

The East Scotland Day Rider is valid as far north as Aberdeen, to Stirling via Perth, Crieff, Pitlochry, Edinburgh and across to Glasgow on the X number Routes (ie not via Edinburgh).

It is a remarkable value ticket. I have used that from Edinburgh to Aberdeen via St Andrews and from Glasgow to St Andrew's thence Edinburgh- 11 hours from Glasgow to Edinburgh with various sightseeing stops.

All these routes are essentially citybuses (albeit the X routes are run with long distance toilet equipped, wi fi and leather seats coach-buses). You can purchase the tickets ahead of time on your 'phone but it gives no priority boarding. They have sufficient capacity for traffic demands.

The X60 is essentially duplicated by the double decker #95 at 1525 and 1635 on a slightly different route (a nicer route between Leven and Buckhaven, on the 'low' road) and at 0833 from Anstruther on the Monday.
That 0833 is a school bus which sets all the little horrors (oops, sorry, children) down at the Waid School 2 minutes later. I've been on the reciprocal afternoon school working and it's not hugely full.