I don't know where you are pulling the 1/25 from. The chance of someone getting gold initiative under the pre-condition of getting a gold somethingis 1/5 (well, slightly above that as two-at-same-time are possible). I don't really know what the probability of getting gold is, but that's hardly a matter of concern here as the OP is not interested in non-gold rolls.

The chance of someone getting 6 gold initiative w/o any other gold is 1/15625 -- something that is probably possible between all the servers.

However, the OP did not explicitly state that the player have no other gold. If, for example, he have a total of 10 gold with 6 init, which is still a significant majority to be of "concern", the chance is actually 0.637%. A small number to be sure, but not nearly small enough to not happen, in fact there would probably be dozens of those.

At the same time, the chance of player getting 5 gold wi/o init is 32.7%. A minority yes, but hardly surprising.

So nothing the OP stated shows any problem. He'd have to present much more complete data and at that mitigate self-selection bias etc to make a meaningful statement.

You are assuming that all number in the range have equal probability, which is not true.

Also, there are 100^5 different outcomes, that's 10 billion. combination of results are multiplicative.

As for number of outcomes with EXACTLY 1 gold, that is 20 x 80^4 x 5, or 4.096 billion.

So if all chances were fair, we'd be getting them 4 out of 10 rolls(more, in fact, as there are chances for multiple gold, but this will suffice), which we obviously don't, ergo the chances are NOT fair, ergo Nothing in your post is correct other than the 1-100 range.