The architect of the 1985 Transport Act, the late Nicholas (later Lord) Ridley, went on record when drawing up the Act as saying he could see no difference between running buses and running sweetshops and believed that having competition would benefit passengers. The reality, as we know, was very different.
Buses are a network-good, but I think of competition as being "within transport" which means that trains, buses, bikes, and cars all compete for "passenger traffic".
Edinburgh does have very minor competition from Borders Buses, McGill's (X38), and Stagecoach which can be a good thing. I like the fact that Borders Buses aren't used much in Edinburgh as it means I can get to Midlothian very quickly. The only real competition is Brightbus which is owned by McGill's but it's for the tourist/visitor market, which is massive in Edinburgh and can probably support 2 operators.
To go back on topic, franchising is a daft idea to run in Edinburgh as this would have the effect of depressing wages for drivers and reducing investment in vehicles and you'd find the big five moving in as it stops being about money in the farebox and starts being about complying with a contract.
I think the core problem in the UK bus industry is not regulation/deregulation, it's the fact that First Group is a publicly listed company, rather than a private one.
First's shareholders clearly don't understand the industry FirstGroup are in as everyone on this forum probably knows of the various bizarre business practices at First, which do make sense to people who went to business school and think bus services are just like business school widgets instead of a network-good. McGill's and West Coast Motors are both private operators, but their services are quite good compared to First, as are most of the larger independent bus companies.
Bringing in franchising would mean that the widget-idiots (or wijiots) would actually be closer to understanding how to run a bus service, but it then becomes about contract design & enforcement, and hiring more expensive lawyers which will only be a bad thing in the long term.