TITLE: Can we prove that there are countably many isomorphism classes of compact Lie groups without the classification of simple Lie algebras?
QUESTION [5 upvotes]: This is an old math.SE question of mine that was never answered:

It is a nontrivial fact that there are only countably many isomorphism classes of compact Lie groups. One can prove this by a series of reductions: first to the connected case, then to the simply connected case, then by classifying simple Lie algebras. Of course, this proof actually gives a much stronger classification result.
If I only want to prove that there are countably many isomorphism classes of compact Lie groups, can I work without appealing to the classification of simple Lie algebras? I have some ideas involving Tannaka's theorem but I haven't worked out a proof yet.

The idea I had was to classify the possible symmetric monoidal [more adjectives if necessary] categories of representations of compact Lie groups; I think these categories are all "finitely presented" in a suitable sense, and from here it should be possible to show that there are only countably many presentations. Such presentations are given for the classical groups in Baez's Higher-Dimensional Algebra II: 2-Hilbert Spaces.

REPLY [2 votes]: In comments it is mentioned that a step in the proof can be " to prove is that the Lie algebras of compact Lie groups are infinitesimally rigid in the sense that they have no first-order deformations". This was first done, without the use of cohomological arguments, in Segal "A class of operator algebras which are determined by groups", Duke Math. Journ. 18, 1951 pages 256-257.
Once one takes for granted rigidity the fact that there are only countably many isomorphism classes of simple Lie algebras seems to me to follow from algebraic arguments. The variety of Lie algebra laws on a vector space of fixed dimension is an algebraic variety, with finitely many components, and each semisimple Lie algebra is a Zariski open dense subset of a component. Thus you have finitely many of them, without relying on classification.
The passage from local to global, then, should be as described by others in comments.