Apparently not:
Check what happened in Spain this season - 4 plus Sevilla (EL winners, outside top 4) and 2 in Europa - both in Group Stages
Because the limit of teams allowed for England is 7 - ordinarily that's 4 CL and 3 Europa. If Man Utd finish outside top 4 and win Europa then it's exceptionally 5 + 2 (effectively this means that if United were outside the top 7 and won the Europa it would become 5 + 2 with "7th" missing out.
But if you look at Spain's entry for EL this season - Athletic Bilbao and Celta Vigo - they both entered in Group stage.
https://kassiesa.home.xs4all.nl/bert/uefa/qual2016.html
Had Sevilla not won the Europa League then Celta Vigo would have gone through the qualifiers. The current information for next season shows this difference with the final place into Q Rd 3
https://kassiesa.home.xs4all.nl/bert/uefa/qual2017.html
It's safe to say the CURRENT situation is:
Champions League
(League winners) 1 as Seed (pot 1, Group Stage)
CL 2 and 3 (ranked, Group stage)
CL 4 4th QR
Europa
FA Cup winners (if Arsenal win cup if not in top 4) - Group stage
League place - Group stage
League Cup winners (currently Man Utd)
Potential Variations:
1. United win Europa, Arsenal win FA Cup (not top 4) - CL becomes 5 with United into GS with only 2 places in Europa (Group stages)
Arsenal if Cup Winners
Next League place(s) eligible
2. United don't win Europa, Chelsea win cup
5th, 6th, 7th basically (the priority for all - where applicable - is Cup winners, 1st league place, 2nd league place, 3rd league place, League Cup winner if used, FA Cup runner-up place (if used).
So if we nab the last place we don't go to QR 3 if United win the Europa. If we finish in the league above the other qualifier (Arsenal if they don't win the FA Cup or United (not winning Europa) we go to the Group stages. It's less likely we'll get 6th but if we do we want Arsenal to lose the FA Cup final. If we finish 7th we want United to win the Europa.