Efficient way to get power levels from an SDR (RTL-SDR specifically)
As far as I understand your question, you want to estimate the PSD in a large bandwidth. 

What you're currently doing is tuning to one part of that bandwidth, receive samples at a low sampling rate, calculate the power in that small bandwidth, retune, and repeat.

So, yes, using a larger sampling rate will obviously reduce the time you need to sense a large bandwidth: A larger sampling rate will give you more spectral information.

The DFT->square method is one classical spectrum estimator. So, yes, instead of tuning $N$ times to receive an $N$th of the overall bandwidth $B$ by sampling at a rate of $\frac BN$, you could just as well sample with a rate of $B$, and calculate $N$-point DFTs in the same time. Since sampling the same time at rate $B$ gives you $N$ times the samples of sampling at $\frac BN$, the information would be the same (Nyquist/Shannon).

You might want to look into other spectral estimators, too – but for now, I don't really see a big disadvantage in simply using the DFT with an appropriate Window – and given the fact that neither the analog nor the digital filters on the RTL dongle probably are overly fantastic, I think you can just stick with a Hann window and be fine.