No, I started with Toro Rosso.
It was the same with last year's game, the team you start out with, even if you don't complete any R&D objectives, will do better after you leave.
I started out with Toro Rosso on JEV's ride and when I left the season after JEV was always making it to Q3 and always finishing within the points. So, I don't know how HRT is doing so well.