Agreed, Ukraine are probably not put off by nukes. However, this is about the west. What do we do after one? After 10? Putin could be running out of non nuke options.
Biden made some sort of promise to Putin that something very unpleasant was going to happen to Russia if they used any nuclear weapons. He may have been somewhat specific in their private conversation, but no further details have been made public. I have seen speculation that the US would lend Ukraine the services of the USAF and USN for a while.
Xi Jingping has also made clear that there would be negative consequences from China if Russia used nuclear weapons. Modi from India has also expressed some hints that the relationship with Russia could get frosty if they decided to use nuclear weapons.
Even if Putin doesn't believe Joe Biden, he probably believe Xi and can't afford to have China join the sanctions against Russia. Russia is completely dependent on China for some materials they need to make weapons. Among them the chemical components for explosives and a number of electronic devices. China is also openly sending non-lethal aid to Russia such as body armor and boots.
China can afford to cut off Russia. Trade with Russia is a tiny percentage of their foreign trade, but Russia would be hurting if China cut them off. At that point Russia would only be able to rely on North Korea and Iran for military aid of any kind.
I think Putin is wise enough to know that using a nuclear weapon would turn out very bad for Russia. However if he is removed and one of the hot heads who don't really have any respect for the international order comes into power, we might see nuclear weapons used.
The world has seen this kind of arrogance before, though not as far as nuclear weapons. In the US there was a think tank in the 90s called the Project for a new American Century (PNAC) who believed that since the US was the only super power left, it should be free to do what it wants in the world and that was a good thing. These people heavily populated the George W Bush administration (2001-2009) and they are responsible for the invasion of Iraq. Iraq was supposed to be the first stop on a remaking of the Middle East.
It was a failure because they really didn't understand how military operations work and that while the US was the strongest force in the world in conventional combat, an occupying force can't stop an insurgency if it is allowed to get started. Between not bringing enough troops and complete incompetence from the occupying administration, the occupation was doomed a few months in.
Russia is guilty of some of that same sort of hubris starting this war, but Putin does know that there are lines he should never cross because getting into a conventional fight with the US doesn't turn out well for the other guy. There are people to Putin's right who believe the only reason they are losing in Ukraine is because they didn't use everything at Russia's disposal including full mobilization and nuclear weapons. If one of these clowns gets into power, he might try both of those and neither is going to go well. Putin has been operating on the world stage for 20 years and he has some ideas what will trigger the west, the others don't have the international experience Putin has and many have the arrogance that natural Russian superiority in all things (that's their belief, reality is different) will win the day.