Obviously companies would like very much to block your right to fix your own products, or force you to take service only from their authorized dealers. Fortunately, thanks to the auto industry, this is a well-trod area of law, although as "products" and "services" keep getting closer and closer together, manufacturers just keep trying. Automakers too - just look at Rich Rebuilds on Youtube, and his struggle with Right to Repair on Tesla cars.
Again fortunately, the political sense in Washington right now, and in California generally, is to strongly protect Right to Repair.
- It will be very difficult for manufacturers to make it illegal for you to fix it.
- Third party repairmen cannot get in legal trouble. (though, manufacturers can hassle them in petty ways; again see Rich Rebuilds.)
However, this doesn't help a lot with your case.
Let's look at your list of recourses.
Warranties do not apply to user-caused damage. For instance if you crash your car, Ford does not cover that under warranty. That is insurance. Similarly, you broke your screen, which is not a factory defect, so it's a matter of insurance not warranty. As such it's not subject to the Magnuson-Moss Warranty Act.
This gets confusing because it's often bundled.
Apple had every right to refuse a warranty repair as it was not a factory defect.
Apple has the right to set any condition on their insurance contract which is not unreasonable, and you can read the contract and see. And in particular things which increase their likelihood of a claim. For instance, that battery may have been thicker than a stock battery, possibly contributing to the screen breakage. Your recourse is to decline that insurance coverage, and get it from someone else.
As far as taking cash for an off-insurance repair, where Apple is playing the role of any random repairman... like any business Apple has the right to refuse service to anyone. *
So yes, Apple can just do that.
* barring certain prohibited categories, like race, religion, national origin and the like.
The financier also gets a say.
Note that if you're not paying $1000-ish upfront for a nice iPhone), there's some sort of financing going on behind the scenes, with the phone as security. Your rights to modify it are limited because you don't own it free and clear. This is often shrouded in smoke and mirrors, like the "free phone" you get in exchange for a contract commitment. Not free, and not yours (yet)... read the contract.