Have England missed a trick by not selecting their best wicket keeper in the first test of the Ashes.

I understand Bairstow runs are important and he should be in the team but at what cost if he has the gloves on. Isn't Foakes a much better wicket keeper?

Foakes is head and shoulders above him in my opinion. I do believe these days a keeper has to score runs.To what cost, that will be decided in the aftermath of a loss.
I love watching a good keeper go about his /her work . If you don’t notice him/her they’ve done the job.
In conclusion, either Foakes gets better with the bat, or Bairstow gets better with the gloves..

Ben Foakes is the best wicket keeper in England and in my opinion should be in their test team. In 20 Tests he averages 32 including 2 centuries and 4 half centuries so he can bat.
Missed opportunities with the gloves is not just about the runs and partnerships the batter goes onto to build, they can deflate the team if they're occurring on a regular basis.
I'd have Bairstow on the team as a batter.

It is simple. In Test Cricket, you can't win if you can't take 20 Wickets.

It's why, in Test Cricket, I would have continued to have picked Ian Healy over Adam Gilchrist. No matter how damaging Gilchrist could be with the bat, Healy, as a pure keeper, was the best that Australia ever produced.

I think England have definitely missed a trick with picking Bairstow over Foakes for the Ashes.

Thanks for the feedback. I think they have missed the trick as by my count, he's missed 4 chances leading into the last day.