A Newry - Belfast bus doesn't provide competition to Translink's Newry - Belfast bus and train services, now you are just being silly.
Trains yes but in the grand scheme of things, Eamon Rooney is running 3 trips per day. Hardly going to hit Translinks bank balance is it!
Derry - Belfast International and Belfast City airports had no direct link from Translink, only connections in Belfast so a private operator saw a gap and started a route, sounds like competition to me.
Not really competition as it would be seen to regulators. Especially Derry to Belfast International, that wasn't covered at all in any meaningful way by Translink. Over to Belfast City, could be some competition but (based off the website), it seems that the smaller stops en route are only if you have booked them online (it says on request but how can you request the bus to stop en route when you don't know when it's due?) It's very minimal competition and again, the amount of potential passengers means that Translink are pretty safe that they won't steal too many passengers and they have a main market that they are focusing on and that isn't normal travellers, it's Airport travellers. An area which Translink aren't really focussed on.
Bus Eireann do not run any joint services with Translink any longer, they withdrew from Derry - Dublin years ago and stopped their share of Belfast - Dublin during the Pandemic. I don't know if they even accept each other's tickets any more on the Strabane-Omagh-Monaghan-Dublin corridor that both Translink and Bus Eireann run services on. John Mc Ginley coaches also run services competing with both Translink and Bus Eireann on that corridor to both Derry and Letterkenny.
As far as I am aware only the (formerly 3) Dublin - Belfast operations are using the EU rule to run without a seperate licence in either country, all other cross-border routes have pick-up and drop off in at least 1 if not both countries.
My mistake, I clearly saw an old timetable. John Mc Ginley, while they do compete in part with Translink (on the Letterkenny to Dublin route), they do serve other markets too and run at different times to Translink. The, on average, 3 trips per day (The same as that provided by Translink), again isn't going to break the bank. The competition is minimal. The Inishowen to Dublin route has only 2 stops within Northern Ireland, those 2 stops being in Derry itself. If John McGinley were to put restrictions in place to prevent local travel here, it should only be a cross border only service in terms of how it is licenced in Northern Ireland (Republic of Ireland it would come under their full licencing as it has local journeys that side of the border)
Bus Eireann competition is a bit different as the Republic of Ireland does permit direct competition, it just heavily restricts it.
I really don't know what point you are even trying to make, companies can set up services in competition to Translink and as I have shown there are some, the opportunities to run a profitable bus service in NI is going to be pretty slim.
They serve markets which Translink aren't interested in or would have minimal effect on Translinks bank balance. If someone tried a Derry to Belfast route though, that wouldn't happen would it? To basically say that the opportunities are slim but there could be some opportunities if it wasn't so locked down to competition.