While it technically doesn't hurt them to be on chick feed, it's formulated for the nutritional needs for rapid development. Once they're grown, some things need to be dialed back, like iron and protein, and others, like calcium and fiber, need to be increased....
If you don't want to get that deep into formulating chick feed with supplements and for seasonality, something like Flock Raiser is a good option
Have you read the labels on the products you are talking about?
You say laying hens should have less protein than is found in chick starter, and you recommend flock raiser.
But according to the Purina website, their chick starter has 18% protein, and their Flock Raiser has 20% protein. So what you recommended has MORE protein, not less.
Yes, laying hens need more calcium, so hens eating chick starter or flock raiser need oyster shell available in a separate dish.
The bag labels do not list iron, but egg yolks contain a lot of iron. So a hen that is laying an egg each day would also need a lot of iron.
I think chick starter is fine for laying hens, if they have a separate calcium source (like oyster shell.) Flock Raiser should be fine too, but chick starter should not cause any problems.