I'm relatively new to posting on here, but I've been reading this thread from time to time, and feel like wading in with my opinion. In my opinion, the idea of London's bus service being run by one operator would lead to even more problems than we have now.
There are ultimately 4 ways of operating services in London:
1. A full monopoly run by Transport for London in house. This would be the same as was between 1933 & the mid 1980s. There could be some cost savings here, and many are in favour of public services being run by the public sector (no public money going into private hands). Whilst many do look back at these times with rose-tinted nostalgia, London Transport did have huge problems throughout these times, ranging from constant staffing issues to large fleets of unreliable buses to plummeting passenger numbers and spiralling costs. The bigger the organisation, the more these issues impact across the network; and those that suffer the most are the fare paying passengers. The political solution here was privatisation, although this isn't without flaws.
2. Deregulation. This is what we have in the rest of the country. Buses are a commercial enterprise, if a route doesn't make a profit, withdraw it unless the local council want to stump up some money for it. If a route is popular, then any operator can have a go. A bus war being fought out on London's streets would be a nightmare, and again we wouldn't have a satisfied public. No integrated ticketing system, fares going up at the whim of the operator and routes constantly changing to meet commercial demand.
3. The current system. This is probably the best of both the above. Operators are rewarded for good performance by keeping and gaining contracts; punished for bad performance by losing contracts. The operators need to make a profit and they cannot charge ridiculous prices as they'll lose out to a lower bidder, the only incentive they have is to run the service as told to. A diverse set of operators means diverse fleets (so if one type of vehicle suffers reliability issues, replacements can be sought quickly). Operators can find it a lot easier to get of poor drivers through redundancy if contracts are lost, and can set competitive pay rates to attract better drivers. The passengers get routes and fares decided by TFL, the bus companies get paid to provide the service. Obviously, there are imperfections in the monitoring of performance, but the principle behind this is probably the best option.
4. The option suggested about handing it all to one company. In my opinion, this the worst of all. Route not running reliably? Tough. Bus driver feels they aren't getting paid enough? Tough, he can't go to another operator. You essentially get all the problems London Transport had, with all the money going to a private operator who can charge whatever they like as they'll have the monopoly.
That's my brief opinion on who should be running services. On fares and money etc, I think TFL were banking on Crossrail to get the books back in shape, but I really do think something has to be done in short term just to get some sort of revenue in. Probably showing my age a bit here, but back in my day, a child fare was just 40p compared with £1 for adults, with child bus passes priced at £4 a week (this was for an all-zones pass). However, the local boroughs also provided free bus passes to those who received free school meals. Child fares set at 75p (with means-tested free fares for those who cannot afford it) would not be so unreasonable, I think its only fair that those using public transport make a financial contribution towards the running cost of the service.