There is something fundamentally wrong with this argument, but I've had to think about it for a minute to come up with a way to explain it.
There is no practical difference between throwing 6 dice together and having them all come up 6 and throwing a single die 6 times and having it come up 6 each time.
Ultimately you've got 6 independent die rolls and coming up with 6 on each. So the formula you were using with 2 dice and 3 dice would also apply to the 6 whether you threw them together or in sequence.
To put it another way, you are correct that for any single throw of the die, one has a 1 in 6 chance of getting a 6. To do that six time in a row is quite different. For instance, for the 6th throw of the die to count if it came up 6, every previous roll would have had to come up 6 but there were 5 chances of 5 in 6 odds against that happening.