Exercise has a massive impact on the health of a chicken and most other creatures I believe. It does for humans. Most of us don't do enough and the few fitness fanatics I know, do too much.
Chickens are class A foragers. Dig here, snatch there, see a spider on the next bush and dive on it, digging, lots of that. They sort of fly as well. It can take a big bird a couple of metres or more to get off the ground on the run. These things are what make a chicken a chicken.

I also believe that chickens that get plenty of exercise and plenty of space to do it in have fewer mental health problems, they just get along better or at least aren't in each others faces all the time.
So, yes I think good health will play a role in the number and severity of egg laying problems but it is worth bearing in mind that for hens dying of old age (9 to 11 years old) reproductive cancers etc are common causes.
My understanding of the primary cause of egg laying problems is the hen isn't built to deal with the high laying rates many now have.
I have had free range hens, not many, over the years that have become egg bound and fortunately I've seen it quickly and dealt with it. Miss that for a few days and you've got a dead hen. Do many people check to see if this it what killed their hen, I doubt it.
I agree with Ridgerunner, too much easy food isn't good for them and while high protein feeds are popular I've known chickens that have done fine on 12% if they been able to forage.
What I have my doubts about is whether a chicken can get enough exercise in a conventional coop and run, no matter what the run contains to make a difference to the number of egg laying problems compared to a free range hen.