Paint does not show true color even after 2 coats on top of existing light color

There's a few things that could be going on here. First, the wall paint could be absorbing the new paint more than the fresh white board or paper you are using. The texture of the wall could make the color appear to be lighter, but not sure if that's the case if you have a flat paint (not glossy). Also, have you ever seen those optical illusions where there are two squares that are the exact same color grey but one looks almost black and the other looks almost white because of the colors that surround it (square A and B are the exact same color)?

The grey on a stark white background is going to seem darker even if there is just a little white border.

So, you have a few choices. You can just accept that it looks lighter in the room and use a darker shade, but you also have to realize that once all the wall is painted, the color will look different than it does now. You can prime and then paint. What I like to do is use a tinted primer then one coat of paint. You get the best of both worlds and typically the one coat (of quality paint) is enough.

Now, as to why your painter wants to use Sherwin Williams paint... It's most likely that is what he is used to. Painters have big brand loyalty, and the paint stores feed this with quantity discounts and "pro accounts" that get lower prices. Also, note that "ProMar 200" sounds like some nice "professional' product, but it's one of the least expensive paints SW sells. It is their value line paint for large volume painters. Home Depot also has a "pro" line of paint, and in my experience "Pro" in the paint world means "good paint for a low price for large jobs". It doesn't mean its somehow better than "consumer" paints.

Direct answer to your two points:

1. If the paint has a hard time covering with the light color you have chosen, it's not a very good paint. Light colors should hide the best because the white pigment in the paint is very strong. Darker colors like reds and blues are notorious for poor coverage, but not grey.

2. Many other paints would probably cover better. ProMar is a base-level paint that works good in most cases, but there are certainly better covering options. Does $5-10/gallon more for a better paint really matter in the long run and in comparison to what you're paying for labor?

Also: specifically not trying to recommend a brand or throw SW under the bus... SW has 4-5 lines of interior paint and I'm sure they have great stuff. I have friends that won't use anything other that Benjamin Moore and others that only buy from Home Depot. I happen to believe that most suppliers have good products and you have to choose the right one based on the job. You can't just blindly use Product X for every single job.