I'm near certain that infrastructure investments from Edinburgh City Council vastly supports lothian buses, from bus stop information and the likely hood that lothian seem to gather much work from thrid parties, such as operating the x98 special service from queensferry including the likes of private hire contract from school, colleges etc around Edinburgh.
Nothing stops Edinburgh Council making shareholder investments towards Lothian Buses either.
As mentioned by others above, infrastructure investments from any council will support the incumbent operator/operators (that is the nature of incumbency), and one would hope that the council and likely operators are in some degree of dialogue before around any major works, tax payers and council residents would have a serious reason for complaint if otherwise.
When it comes to tenders or contracts (and I admit this is not my area of expertise), but I am pretty sure that the city council would have to in the large majority of circumstances put the contracts out to open tender or risk falling foul of competition and state aid laws. Where Lothian benefit in this is that they are large enough to post the bonds/guarantees that smaller operators may not manage, and have benefit of depots and fleet in the right places to service Edinburgh and the Lothians.
Given the prices charged on most specials, I highly doubt they are done on a contract basis with the city council, however they may be contracted by 3rd parties. Obviously then it is simply Lothian benefiting from the incumbency factor and good corporate relationships.
I had a quick look at Lothian Buses Group financials from 2013 onward (that gives a 5 year window and is largely post the Transport for Edinburgh restructuring).
In that time there have been no capital contributions from any of Lothian Buses shareholders (Share Capital has remained at GBP 6.4m). Lothian Buses have however generated a net profit of GBP 42.6m and distributed dividends of GBP 27.7m in that period. Lothian have also sold goods or services to Edinburgh (including Transport for Edinburgh (TFE) and Edinburgh Trams), Midlothian and East Lothian Councils worth GBP 16.1m while purchasing GBP 7.3m. As such the councils have seen a net benefit of GBP 18.8m (there is a rounding difference) over the 5 years, arguably the opposite of funding. (I realise there is a slight complication in that Edinburgh councils share of dividends actually flow to TFE, but the point still holds)
Interestingly in the same period TFE group (this includes Lothian Buses as above) had a GBP 5.8m Capital investment in 2013 (although I this was really just the restructuring) but has no change since then. They have generated net profit of GBP 43.8m and distributed GBP 18.9 m in dividends to Edinburgh council (I have not included the non-controlling interests distribution as this will just be the 3 Lothian councils share of Lothian Buses). TFE Sold goods and services worth GBP 10.2m and purchased goods and services worth 20.5m in the same 5 year period from Edinburgh Council. As such Edinburgh Council had a net benefit of GBP 29.2m (again a rounding difference). TFE is however made more complex as they include Edinburgh Trams, so it isn't just the buses.