I don't know how true that is.
It is true that New Street (fantastic building as it was) in Edinburgh had some challenging issues given its location and internal clearances. Again, will need to refer to the experts e.g.
@DunsBus whether it was this that precluded new Ailsas being allocated there. Not certain where the secondhand ones were allocated for city services.
However, there were plenty of other depots that could have had Ailsas but didn't. From what I can recall, during the 1970s, Eastern standardised on the low height ECW bodied Fleetline, 5 batches during the 1970s as well as getting some ex Central ones. Eastern then experienced a major vehicle shortage (hey, it was the 70s). So it gained 10 Ailsas at Musselburgh, diverted from Fife in 1977, before resuming Fleetlines that were dispersed across the network.
With the end of Fleetline production, they had a choice to buy Ailsas or something else. They dual sourced with Ailsas for other mid-Lothian depots at Musselburgh and Dalkeith. However, other depots (aside from New St) with deckers received Olympians with ECW and Alexander bodies, namely Livingston, Bathgate, Baillieston and Dunbar - don't know if any had height issues though I doubt the first three had.
As an aside, all of these huge NBC and SBG monoliths had these disparities in the extent of their territory but still find it sort of odd that you had one single firm with services and a depot in Glasgow, and in Edinburgh, plus places like Jedburgh, Kelso and Berwick. Mind you, they shared the latter with a firm who also had a depot in Scarborough!