The Black Sea ports were the most important export ports for the Russian Empire, not only due to their proximity to the Ukrainian fields that grew the wheat Russia exported, but also to the rail network necessary to carry those products to the ports. You can see a map of Russian rail infrastructure in 1916 here. As you can see, the overwhelming majority of Russian railways are in European Russia and Western European Russia at that. While that seems to leave the prospect of redirecting exports through the Baltic ports, don’t forget that Russia is at war with Germany as well, and the Imperial German Navy is a much bigger threat than the Ottoman Navy. Any ship traffic through the Baltic would have to pass perilously close to German bases and the German fleet could relatively easily relocate via the Kiel Canal. While Jackie Fisher did have schemes of launching major offensive into the Baltic to link up with Russia, these never really progressed beyond being just schemes.

The needs of rail transport help to exclude most of the other ports you mention. Murmansk and other ports on the White Sea have relatively limited rail connections to the rest of Russia, even ignoring the weather difficulties in trying to ship through the Arctic. Similar problems apply to the Russian Far East, where there are really on a small handful of ports of note. There was almost no infrastructure along the Sea of Ohtohsk or the Bering Sea. While Vladivostok was connected to European Russia via the Trans-Siberian Railway, when combined with the additional distance, shipments through Russian Pacific ports could have hardly made up for the lack of access to the key European ports.