I had a dealer threaten that it would void my warranty if I did not have them put nitrogen in my tires. I told him that if it made him feel better he could do it for free. He did. As soon as I got home I took the stupid looking green valve stem caps off and put proper black ones back on. Putting pure nitrogen in your tires will never hurt anything, but it will not help anything either.
One big point that most seem to miss including in the vid above. is that the tire is already filled with air BEFORE you start to add nitrogen to the tire. NO ONE pulls a vacuum on the tire to remove the air before adding nitrogen. So there will still be Oxygen and all of the other gasses in there.
But to be honest, there are a very few very specialized places where nitrogen in a tire does help, but they are very few and passenger car tires are not one of them.
Helium is a very small atom, it is hard to keep in air / nitrogen is a much bigger atom so it needs a much bigger hole to leak out of. Heck they could make a better argument if they were filling the tires with Argon gas. Argon is also cheap and readily available and has similar chemical and physical properties to nitrogen but is an even bigger atom to help reduce leakage.
When I was racing a stock car we always put nitrogen in the tires, not for any of the supposed performance claims, but it was convenient. It was much easier to have a bottle of nitrogen in the back of the truck to run all of the air tools than to have a huge generator that could start a compressor. Race tires leak thru the sidewall like crazy. If you inflated a tire to 32psi and sprayed soapy water on the sidewall it would turn into a big doughnut of foam. One nitrogen bottle wold last about 4 weeks of racing using it at the track for tire changes and filling tires and any other air tools we might need to use. and it is way quieter than a generator and a compressor.