Broad gauge carriages and locos were apparently little or no larger than standard gauge. The wheels were still within roughly the same overall body width, the carriages were never ‘scaled up’, indeed if they had been, any alleged greater stability would have been gradually reduced...
The supposed advantage of the broad gauge was that it would allow locomotive boilers to sit between wheels, rather than over them, thereby lowering the centre of gravity and improving stability. In response, the LNWR argued that stability wasn't the limiting factor on speed - it was the quality of the tracklaying.
In the later days of broad gauge, the stock was basically just standard gauge bodies with unreasonably wide running boards, since everyone knew it would be converted sooner or later. Removing running boards and swapping bogies was much easier than building new bodies.
Track gauge has nothing to do with rolling stock dimensions - look at the locos and trains in South Africa, larger than anything in Britain but on 3'6" gauge. Or the vast double-stacks in the USA, way beyond what old Hastings Size rolling stock was restricted to in Britain, but on the same gauge tracks.
Coming the other way - the standard North American loading gauge, AAR Plate B, is about the same size as the loading gauge on the GWR's
broad gauge lines. Slightly narrower, but higher and with a fatter top. And AAR Plate B is their minimum gauge, exceeded on virtually all lines.
The BART trains in San Francisco had every design aspect in the 1960s gone into by "experts", so they ended up with non-standard 5'6" track gauge, cars no larger than the rest of the USA because the air resistance expert pointed to minimising piston pressure effect in the tunnels, and the inability to be able to use any of the standardised track maintenance equipment from the various US contractors, so they have to buy their own specialist kit for everything. They really regret going for a silly gauge.
In a particular stroke of genius, the 'experts' were from outside the railway industry, and insisted on doing everything from scratch. Whatever the theoretical advantages of the 5'6" gauge may have been, they didn't understand the rail-wheel interface and specified cylindrical, rather than coned, wheels. As a result, BART has major issues with flange wear.