I'm a bit skeptical as to how simple and low-tech a proof can be, since a fair amount of Lie group theory has to be in hand. Short of going into the full details of Iwasawa decomposition, you might find it worthwhile to look at two old papers in the Bulletin of the AMS. One is by Mostow:
- A new proof of E. Cartan's theorem on the topology of semi-simple groups, Bull. Amer. Math. Soc. 55 (1949), 969-980, doi:10.1090/S0002-9904-1949-09325-4,
giving a new proof of Cartan's theorem, while the other is a longer survey on the topology of Lie groups by Samelson:
- Topology of Lie groups, Bull. Amer. Math. Soc. 58 (1952), 2-37, doi:10.1090/S0002-9904-1952-09544-6.
Concerning the further question about conjugacy of maximal compact subgroups, this strikes me as too deeply enmeshed in the structure theory of Lie groups to be proved easily or quickly. I'd want to start by looking carefully at how this conjugacy theorem has been approached over the years (but of course without ruling out a better approach).
