Just "cut" in a random location (rotate the deck a random amount), then split the deck roughly in half (add a bit of random variation), then "flip them together" back to front by alternately taking 1 or 2 (randomly, add a small chance of 3, so maybe 50% 1, 40% 2 and 10% 3) from each side till there are no cards left to shuffle. Then repeat 8 times or so (there's a certain minimum number of times that ensures good randomness)
For the first problem, a very simple "for each card, pick a number between 1 and 52 and swap the card with that number (or do nothing if it's the same card)" is proposed to eliminate bias. This seems pretty simple to verify.
For the second problem, there is no workaround except to use a better RNG.
So in the context of the paper, the reason your algorithm wouldn't have worked is because the programmers seeded their PRNG with the time of day. In the context of shuffling more generally, I don't know but it seems that there are even simpler algorithms.
That paper: https://web.archive.org/web/20140104095330/http:/www.cigital...