>>If Putin goes nuclear, Russia gets erased in a full exchange.

If it goes intercontinental, then absolutely - along with wherever their missile submarines can strike back. After that, it's nuclear winter for most of the planet, the 'global order' goes out the window so even places untouched in the Southern Hemisphere that have soils that need industrially produced amendments to be productive (e.g. Brazil, most sub-Saharan African nations) lose those inputs and go back to earlier productivity levels - even after the 'winter' (best case, 2 years like after Krakatoa) is over. At a minimum multiple billions of people die, some from the nuclear exchange, the majority from starvation. A lot more than ‘Russia erased’.

I suspect the Western response will be much more moderate, probably an overwhelming conventional one rather than anything nuclear. Clearing the seas of the Russian surface navy is a given — and Finland can easily disrupt any shipments to Murmansk (big submarine base).

Destroying Russia's oil/gas wells or internal pipelines are probably a better target for a conventional response rather than any population centers. Russia's been using winter as a weapon since forever. Step one would be to warn the Chinese that the Siberian wells are also a potential target.

After that, it's the same nuclear standoff. Russian population centers are still there as 'hostage' against sub/nuke retaliation.