If the "adult feed" is a layer feed that contains high calcium, the chicks would do better on a lower-calcium feed anyway. Molting hens don't need the extra calcium either, and I already noticed that you do provide oyster shell so they can help themselves when they need it.
As regards free ranging and getting enough protein: it depends on what they find while ranging. They might be feasting on bugs and worms, or they might not find any. So they might be fine.
But overall, your chickens, your choice. The higher calcium isn't going to make the chicks drop dead in the short term, and the protein they do or don't get from free ranging isn't going to be a life-and-death matter for them anyway.
Is the high protein feed that much more expensive? Cat or dog kibble is usually more expensive on a per-pound basis than any chicken feed. And do read the label. If the dog kibble is 20% protein and chick starter is 20% protein, and the dog kibble is more expensive, then you do not save money that way. (20% protein is a fairly common number for both dog kibble and chick starter, but check you labels to know what you are actually dealing with.)
As a protein supplement, it might work. As a money-saving measure, I expect it to be a total failure. It is almost always cheaper to buy a higher protein chicken feed than to buy the low protein feed and high protein treats. Treats tend to be badly overpriced for the amount of nutrition in them.
Here are some exact numbers:
https://www.tractorsupply.com/tsc/product/manna-pro-flock-party-merry-minnows-10-oz-1030177?
Minnow treats at Tractor Supply company, $12.99 for 10 ounces at 55% percent protein
That works out to .34 pounds of actual protein in the bag, or about 1/3 pound, currently priced at $12.99
So to get one pound of protein would take 3 bags, at a total cost of $38.97
For any chicken feed that comes in 50 pound sacks:
2% higher protein means 1 more pound of actual protein in the bag.
Does a bag of 18% protein feed cost $38.97 more than a bag of 16% protein feed? Or a bag of 20% protein feed cost $77.94 more than a bag of 16% protein feed?
(50 pound sacks of feed are common, and make the numbers easy. Other sizes will show similar benefits, when compared with almost anything marketed as a treat.)