A solution is often just getting rid of the problem. No third party = no problem. The real problem that surfaces then is probably that it's too profitable to throw out and so it becomes something Amazon just chooses to live with as long as it can.
The problem is they built it into the main UI with no care for distinction or customer appraisal of what is going on. And now they have this mess.
They also have a bunch of their own brands: https://www.amazon.com/b?ie=UTF8&node=17728530011
That said though - this would clearly impact profits proportional to the level of strictness/barriers.
Amazon's whole business is being the fulfillment for 3rd party sellers, you take away that you take away the whole reason people use it. If Amazon was just online 1st party Walmart they would sell almost nothing.
Amazon made the decision to pivot to third party sellers because they realized how poorly being an actual online merchant scaled. Why get into that game when you can be the platform and vertically integrate everything from the listing to last mile delivery charging a fee every step of the way?