I can't imagine that multiple coats of paint would be a problem unless the painters are very bad.

Two coats of primer?!!??
Almost NEVER needed.

CLEAN walls are the most important item.
THEN priming...ONE full coat.

Scuff-sanding existing paint only becomes necessary if the paint is a Satin or higher sheen. Regretably, this step is often just "blown off", as is washing walls b4 priming/painting!

Yes...ideally...2 topcoats on the primecoat.

Faron

"I'm worried about there being too much paint on the walls if they do 1-2 coats of primer and then 2 coats of the Versatile Gray."

Too much paint? No one sands the paint off when they repaint, unless it's a decades-old house with 50 or 60 repaint jobs.

If the current color was put on over a primer, and it's only a shade or two off, they shouldn't need to prime, just repaint the correct color.

There are walls in my house that I have painted 10 times in 5 years... yes, I know I have a problem!

You do not have a problem. Actually, it's good. The first coat kind of is absorbed in, the second coat which you might be doing down the road is hard to roll on because of the new plaster etc.

So, another coat of primer and paint is good! I assume you are not paying, correct??