Some operators who did introduce fareboxes when urban OPO became a thing found that overall takings didn't match what they should have been, as drivers weren't too bothered what went in to the box, as no individual driver could be held accountable for what any bus ended up with at the end of the day.
Broadly correct. Many urban operators went for fareboxes simply because when they moved to One Person Operation OPO (or as it was at the time OMO - most staff were men), then there was a view that it would speed up boarding and so the loss of conductors wouldn't be felt so much. There was also the reduction on the amount of extra work placed on drivers, and the security aspect; not so much the issue of fiddling (which could happen with conductors) but that instead of the relative safety of a two man crew, you would have one person who could be robbed.
My first experience of farebox operation was in Birmingham in the early'70s. I think you are right, it was about the speed of boarding. not that the driver's couldn't be bothered. Rarely passengers could ask others whether they had change; otherwise, if they didn't have enough, it might be a question of 'pay what you've got' (although I suspect drivers knew those who habitually tried to underpay). Also, many's the time I witnessed the pile-on during the evening peak in Navigation Street or John Bright Street, the numbers waiting to board were quite often more than would fit on the bus (and many more than were supposed to fit on the bus). There is no way that the driver could check the amount put in the hopper for everyone that got on. There was a pretty good inspection programme, several inspectors boarded at one time, sometimes supported by Her Majesty's Constabulary but I can't remember these really taking place during the peaks.
One reason why farebox operation was a decent solution was that the fares were low, the max would be somewhere in the region of 20p in the early days. A driver could see whether the amount was right or about right in seconds. Now, unless the company has a policy of very few different fares (on the route I used to use, 45 minutes end to end, there are two for adults), checking the fare if the passenger has raided the money box to find change might take as long or even longer than if the had offered a five pound note, and the driver had had to give change.
With card payment becoming more prevalent and a steady number of passengers using passes (paid or unpaid), exact fare is probably not so important.
A tangent: 'most staff were men'. I remember there were just two women drivers on the Pershore Road routes. I think the first women drivers I had come across. One has exceptional, guaranteed smooth ride (particularly useful if you were standing), you'd be at the destination before you knew it. If I had any doubts about the ability of women to drive buses, they were dispelled.
I'd like the exact fare policy to become universal, just what happens in Hong Kong. It's so annoying to have the driver count changes because it takes so much time, that a single passenger can consume over 20 seconds at the door and delay the whole bus. Hong Kong has already had this policy since the 70s in order to get rid of the conductor on buses (previously fare was paid to a conductor), and by law the fare has to be clearly shown at the farebox.
London Buses are the best in the country (as demonstrated by the ridership) and I believe the whole country should learn from their operation, including two-door buses, headway-based scheduling, etc.
If they had been introduced years ago, when many more people used the bus, it would probably have been a good idea but now, we are trying to get as many as possible to use public transport, we probably need as few obstacles as possible.
Is it a single fare, for any distance? Once you rack up 6 or 7 price points, where the potential passenger does not know the fare when they get on so has to find, say, £5.70, when their 'usual' journey is £4.60, you will soon eat up 20 seconds. If it is a flat fare (or a short hop and one other fare), getting the fare right so those that travel a fairly short distance don't feel they are subsidising those travelling some way, particularly if there are few of the latter because much of the route is rural, is a difficult balancing act.