Proving Matrix Lie Groups are Lie Groups without using Lie Algebras

Question

As the title suggests, I'm wondering if there are any proofs that Matrix Lie Groups (defined as closed subgroups of $GL_{n}(\mathbb{C})$) are Lie Groups that do not require the use of Lie algebras. I'm aware of the general proofs owed to von Neumann and Cartan, however, these require Lie algebras and I would like to avoid using that due to the scope of the project I am working on. I have been advised to look at "Lessons of differentiable geometry" by Postnikov. However, I cannot find a proof in there that does not require Lie Algebras. Is such a proof possible?