Buryatia (not just modern Buryatia, I mean the entire region around Lake Baikal) is perhaps the second most viable East Siberian state, but without Russian interference it would likely just be considered to be North Mongolia. Even if Mongolia ends up a dependency of another country I think the suzerain would back Mongolia's claim to the region, even if it's just out of land-hunger. Likewise, I think the arc from Chita to Khabarovsk to Vladivostok would be considered an extension of Manchuria.
If you want to keep Russia out of Siberia you'd absolutely need a militarized state blocking their path; the Sibir Khanate was almost this but then Russian migrants pretty much dismantled it from within. Maybe a surviving Timurid union of Iran and Central Asia propping up an northward-expanding Golden Horde successor (e.g. Kazakhs) could do the trick? Russia can definitely be held off (the Crimean khanate did this well into the 1700s) but it'll take work. The main problem is that Siberia's population density is quite low and there's few ways to naturally boost it given how hard agriculture is, but a state with an economic/population base elsewhere (Kazakhstan, Oirat lands) has a chance of offsetting those disadvantages. Hey, maybe instead of the Kalmyks migrating straight-west you could have them go kinda northwest and bring other Oirats with them to the Ob valley, creating a northern Dzungaria.
Another possible scenario is a Zaporizhian-Host analogue where you have a state of Slavs (even Cossacks) but they see the Tsar as a threat to their lifestyle and accept patronage from stronger states to hold Russia off. For the Ukrainians it was the Ottomans, maybe this "Tobolsk Host" could be backed by the uber-Timurids mentioned earlier. The Tobolsk Host could, however, also seek the patronage of Western powers-- the Pomor port of Mangazeya on the Arctic coast was so popular with English/Swedish fur-traders that it began cutting into the profits of the inland furtrappers in Tyumen, who then petitioned the Tsar to shut Mangazeya down. Maybe a Kalmar Union/mega-Sweden could strike up a healthy correspondence with Tobolsk and guarantee its independence?
The other rather existential threat for a Siberian-majority Siberia would be China and to a lesser extent Korea-- farmers from both countries would likely flee to Siberia in the case of unrest back home, and may create local self-government structures along the lines of the kongsi republics in Borneo or Korean village communes in Gando. This could potentially be turned to the Siberians' favor-- if it's mostly just East-Asian men fleeing up there, they might take local wives and then a couple generations down the line you have a Peranakan group of people who are technically Chinese but see their priorities as locally-rooted and fear Chinese colonization as nothing but trouble for them (it would only bring competitors for Siberia's mining riches). In the absolute most optimistic case you could see these Northern Peranakan adopt Sakha as their main language of daily life and thus assist in any pan-Turkist unification movements (the Indonesian Chinese adoption of Malay helped secure that language's future as the pan-Indonesian tongue). However I don't have much faith in this possibility. In Maritime SEA, the Chinese were filling niches within an existing economy. Malay was already the language of the bazaar, and upon entering the bazaar the Chinese adopted it. In Siberia they'd pretty much be inventing a whole new agricultural/retail/export economy that the locals may (not) participate in that heavily outside of the lower rungs of resource extraction (furtrapping, logging).
Besides the west and south the final area of concern would likely be the eastern coast, though metalinvader has discussed the difficulties of colonizing through there. Still, a European or Asian naval power will definitely try establishing some manner of protectorate over the coast, at which point they'd probably swamp the local Chukchi, Itelmen, and Tungusic peoples. And then as long as the interior peoples are good economic partners I think they'll be content to let the maximum extent of their inland expansion be missionary expeditions. In the best case for Siberia they can serve as a block on a northward expansion of East Asia, similar to the role Russia played for independent Mongolia. Hell, they might even prop up a Sakha
tygyn as "High Lord of the Northern Wastes" in order to keep any western/southern encroachers out and to gain a local ally for their economic exploitation/proselytizing.
I agree that a pan-Turkist movement is perhaps the best best for achieving an all-Siberian state, and as partners for the Sakha I'd suggest the Altai (who developed a
millenarian resistance religion with pan-Turkic elements during the Russian Revolution) and Tuvans. Maybe the state could be called
Uriankhai, it seems to have been a popular catch-all exonym for northern forest Turks. They'd have to come to some accommodation with the Uralic Nenets and Tungusic Evenks, though.
So basically I'm imagining something that kinda looks like this:
Here's another map with the river routes featured

. (Western buffer centered on Ob-Irtysh, Uriankhai on Yenisei-Lena)
For economic prospects with those borders: the Eastern Buffer could survive off fishing, North Mongolia/Manchuria would have their destinies tied to their respecting heartlands, the Western Buffer could make a real killing off Tyumen oil and then phase into Yamal natural-gas a-la-Qatar if literally everything goes right. I think the best case scenario for Uriankhai would be a Finland-style transformation (starting off with forestry, then getting German investors to build local paper mills, and then on and on with bigger and bigger industrial ventures) that lessens the dependence of the economy on mining. For all this positive transformation to happen, the area would have to avoid getting caught up in great-power struggles. It might be able to do that by virtue of its isolation, but that didn't work out so well for South America...
Going back to basics, though, I think the real thorny issue is that due to the indigenes' low population density (although it might be higher in a no-colonization scenario; the Russians did kill a
lot of people on the eastward march) and lack of economic/military clout, the fate of Siberia depends to a large extent on actors outside Siberia. And engineering changes to keep those actors out will itself cause massive changes elsewhere (the impact of tiny Russia alone is world-shaking).