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This is excellent, thank you. I need to check the details of Step 1 myself carefully, but seems obvious: $f_i = \sum_j f_j p_{j,i}$ being $0$ implies $f_j$ is $0$ for all $j \sim i$; iterate. But, Step 3 is less clear to me—both parts, in fact. I'll have a think some more, but might comment here again if I'm still stuck! (if that's ok)
@JochenGlueck Yes, of course, thanks for pointing this out. Not sure what I was thinking! I actually associate aperiodicity with $-1$ eigenvalue more (eg, non-lazy random walk on a cycle). I almost always work with reversible chains in my research, so I often forget about complex eigenvalues! Still, at least all have non-negative real part...