Can't match a TypedDict with a Protocol

Bug Report

I was trying to use a Protocol to fish out the value type from a TypedDict type corresponding to a key Literal passed to a function, so I could bind it to a TypeVar. That didn't work, but in trying to debug that I noticed that it's just straight up not possible to get a TypedDict to match a simple Protocol that we can see at type-checking time it does in fact match.

To Reproduce

See https://mypy-play.net/?mypy=latest&python=3.12&gist=928611fe3fcf77630451b8ca6b26dfca

from typing import Literal, TypedDict, Protocol, Optional

class Foo(TypedDict):
    bar: int

concrete_var: Foo = Foo(bar=5)

# This typechecks (though it has to be optional for some reason even if the key
# is Required in the TypedDict):
int_var: Optional[int] = concrete_var.get("bar")

class Gettable(Protocol):
    def get(self, key: Literal["bar"]) -> Optional[int]: ...

# But we don't use that to know that the protocol matches. This fails:
test_var: Gettable = concrete_var 

Actual Behavior

main.py:16: error: Incompatible types in assignment (expression has type "Foo", variable has type "Gettable")  [assignment]
main.py:16: note: Following member(s) of "Foo" have conflicts:
main.py:16: note:     Expected:
main.py:16: note:         def get(self, key: Literal['bar']) -> int | None
main.py:16: note:     Got:
main.py:16: note:         @overload
main.py:16: note:         def get(self, str, /) -> object
main.py:16: note:         @overload
main.py:16: note:         def [_T] get(self, str, /, default: object) -> object
Found 1 error in 1 file (checked 1 source file)

MyPy appears to have forgotten everything it ever knew about the types of the things in the TypedDict when it tries to match the get() method to the protocol.

Your Environment

• Mypy version used: 1.9.0
• Python version used: 3.10 and 3.12

Start by running the linked mypy-play reproduction with the provided TypedDict and Protocol definitions. Trace how TypedDict get overloads are compared with Protocol methods; done means the reported assignment is correctly accepted, with regression coverage in the relevant existing test area.