Gideon v. Wainwright established that indigent defendants are entitled to free legal representation. There's no such right for non-indigent defendants.

Not even when the price of legal representation is above my means to pay?

If you need an attorney and cannot afford one, one is appointed. If you can afford one, you can't really say "I'd rather not pay that."

In this context "afford" typically means you can still keep your home/apartment and go grocery shopping.

Note that "Cannot pay" is often interpreted to mean "Has already mortgaged the house and liquidated all things", not "has an empty checking account".

States establish indigence standards. Generally if your income exceeds 200% the federal poverty line, you aren't entitled to a court appointed attorney. Even then, if you're above 125% the poverty line, you can be forced to pay for your court appointed attorney. You can claim financial hardship, but the court isn't obliged to accept your claims.

That's why you should be able to chose the court appointed attorney within reason.

You'd need to pretty carefully define "within reason", though; it's not reasonable to select without regard to cost.