As @Ramiro Lafuente pointed out in a [comment](https://mathoverflow.net/questions/378266/a-question-regarding-the-action-of-a-lie-subgroup#comment960592_378266), there's a gap in this proof. 

You're apparently using the first edition of my *Smooth Manifolds* book. The problem is fixed in the second edition, because the closed subgroup theorem is proved before this theorem. 

Meanwhile, some years ago I posted a correction in my errata list for the first edition, which you can find [here][1].


  [1]: https://sites.math.washington.edu/~lee/Books/ISM/