Indeed. The bus-rail markets do overlap occasionally, but it's fairly minimal. On nearly all flows between towns or to and from major centres, either the train or the bus has a clear edge and most people just use on or the other.
Many of the stage carriage routes where there's true competition are often not part of the £2 scheme. For example, Oxford Tube between London and Oxford is very competitive with GWR, Red Arrow between Nottingham and Derby is very competitive with EMR and CrossCountry, and Greenline 757 between Luton or Luton Airport and London is very competitive with EMR and Thameslink, however none of these routes are receiving the fare cap grant.
I can only think of two corridors where there's true competition and the capped fare grant is at play. One is the X43 Burnley - Manchester fast, which is eligible for the £2 fares and is very competitive on time and is triple the frequency of Northern. The other is X15 Newcastle - Morpeth - Alnwick. These are much less competitive on times, but they are much more competitive once one factors in the length of the walk from the railway station to the town centre in Morpeth, and the bus or taxi journey required for this in Alnmouth (walking isn't really an option due to the lack of a footway). Doubtless there will be a small handful of other contenders for this, such as Slough - London, Norwich - Great Yarmouth, Stockport - Buxton, York - Hull and Leeds - Harrogate. But it's overall a short list.
Many stage carriage services in Northern England have effectively ceded the interurban market back to the railway and coach lines recently, such as Manchester - Blackburn, Manchester - Leeds and Manchester - Sheffield. The Leeds - York fast services are also gone Accrington - Manchester has been slowed down so much it's no longer competitive.