It kind of matters but nowhere near as much as your learning, patience, and ability. I find that most people give up too quickly or lose interest. Or they don't take the time to properly educate themselves about parrots. Equal likelihood of failing with any species in that case.
That said, with an equally positive dedication to proper training I think the Conure or Meyers will be a bit better at training. I find that the bigger parrots are a bit smarter but usually too smart for their own good. They tend to overthink things. Also the bigger parrots are slower. This is good and bad. On one hand it is easier to train them because you can take your time more. Little birds are all over the place. But on the other hand, you can achieve results sooner with the smaller ones requiring less patience.
I found that you can teach something like the
Turn Around trick in about 100 times in 1 day to a budgie, 50 times in 2-3 days to a Senegal, 30 times in 3-5 days to a macaw. So the bigger bird learns in less times but because they are bigger it takes longer and you can't do as many times per day or session. So it takes less number of times but more time in human days. Senegal, Meyers, and birds like that are in my opinion the sweet spot of intelligence, speed, and good learning. But like I said, owner patience and technique matter most.
Since you have a budgie, make sure you are successful and experienced at training before considering another bird.