I can't do mathematics at all without drawing and I have practically never given a course or a talk without making coloured doodles on the blackboard.
There are a few exceptions, as when I had to teach the Sylow theorems, but then I had the feeling that I didn't understand what I was talking about, although the proofs were (I hope!) correct in the logical sense.
The same goes for hard analysis, say linear partial differential equations: I tried to read Hörmander but gave up, because I couldn't get a gut feeling for the inequalities there (even though geometry is definitely present in that book).
I am in a field (algebraic geometry) where it is easy to make drawings and I started to understand scheme theory only when I saw Mumford's drawings of $Spec (\mathbb Z)$, $Spec (\mathbb Z[T])$ and his cartoonesque rendition of the spectrum $Spec(\mathcal O_{X,O})$ of the local ring at the origin of the plane $X=Spec(\mathbb A^2_k)$ over the field $k$, where the closed points of curves have disappeared and only their generic point is left behind , exactly like the grin of the Cheshire cat in Alice in Wonderland, the masterpiece of that wonderful mathematician who also loved drawings.
[If you have never seen a live scheme in its natural habitat, look at pages 72-75 of Mumford 's Red Book where the pictures I evoke above are to be found. Or here pages 111-112]
In conclusion, if you feel drawings help you, by all means go ahead: I find your version of a covering of a compact space truly ingenious an illuminating.
And to finish on a lighter note, there is this story of an engineer asking a mathematician how he could visualize 4-space: "Very easy, I imagine $\mathbb R^n$ and I specialize to $n=4$"