that's a bit unrealistic to force someone to roll a dice around 600 times just to generate a single bitcoin private key.
Exactly. Which is part of the reason I am arguing against using dice. If you instead want to test whether a single die has no bias and be reasonably confident in your conclusions, then it would require even more rolls than the ~16,000 coin flips I gave above to test for a coin. Why take the risk, when there are safer, simpler, and quicker methods available?
maybe a way to shortcut that process would be to take 3 dice and roll them all at the same time.
That wouldn't work. You need to decide in advance which die will be your first number, which will be the second, and which will be the third, as if you wait until after you have rolled to pick the order then you introduce bias. In such a scenario, if die 1 has a bias towards 1 and die 2 has a bias towards 2, then ending up with HHH will be more likely than any other combination.
The method only works on a single die because each individual roll has the exact same chance to be biased as every other roll.