You don't need to be in the Premiership to be able to build a Premiership standard stadium, nor do you need a big backer.
Swansea, MK Dons and Hull all built quality stadiums whilst in the lower Leagues. A number of others could be added to the list.
Also Plymouth, Bristol and Nottingham Forest all have plans for large stadiums. Yes, it is for the World Cup but the Plymouth and Bristol ones are happening regardless of a successful 2018 bid and will just be made bigger if England do win.
Also Pompey's spending in the Premiership wasn't at a level to get 17th place. They were spending huge sums on wages and transfers far in excess of a bottom 6 side.
Pompey could easily have rebuilt Fratton Park and stayed in the Premiership during those 7 years. They may not have had top half finishes or an FA Cup win but it would have secured the future of the club and still remained in the Premiership.
Thus not in anyway a catch 22.