"the idea that it would require extra buses or drivers doesn't seem to add up"
The assumption is that to maintain current service with slowed down buses would require more buses.
BUT
And there are a lot of buts -
As buses hardly ever go above 20 - except on main roads where there aren't as many stops (and will probably remain 30 anyway) this suggestion is questionable.
20mph zones may well encourage more people to travel by bus - car journeys will be slightly slower and roads (eg Leith Walk) might feel less like race tracks (might even be fewer cars) and so feel easier/safer to cross.
However bus journey times are affected by a lot more than maximum speed limits -
More people getting on (which presumably they want) - especially if they don't have Oyster type cards...
Bus waiting to get stops because there is another bus in front.
Too much traffic (generally).
Bicycles stuck in tram tracks.
Inappropriately parked cars (often quite legally near difficult bends for instance).
In short LB should actually WELCOME 20mph zones and all other measures that might make Edinburgh a much less car dominated city - and much friendlier to pedestrians and bus passengers.