Nearly all of "Alaskan" gold was actually in the Yukon Territory. In Canada. In the British Empire.
While a relatively few placer deposits were located in Alaska along the current and former beds of the Yukon and it's tributaries, the "mother lodes" which had shed those deposits were located in the Yukon Territory. In Canada. In the British Empire.
Alaska was merely an easier way to get to the Yukon. Prospectors passed through Skagway or up the Yukon river to get to the territory because that was the quickest way to get to fields. The Alaskan gold rush was in all actuality people rushing through Alaska to get to the Yukon.
A gold strike in the Yukon thirty years earlier will find all the parties involved less able to control the situation. Canada is on the brink of confederation and has a very limited presence in the region. Contrast that to the 1890s when the RCMP was able to quickly establish a permanent presence in the Yukon and even enforce provisioning requirements for prospectors traversing the pass at Skagway.
Because Russia's presence in Alaska proper in 1867 is far less than the US presence thirty years later, Russia will most likely lose control of the rush passing through it's Alaskan territories. Prospectors, as they did throughout the 19th Century, will flock to the region and Russia will be unable to monitor, let alone police, their activities.
With the majority of the gold on Canadian soil and Russia unable to control the increasingly anarchic situation, selling off the resulting headache to the US may still occur.