I made the video you're asking about, Alyssa. The reason for injecting air into the air of the vial is because you're replacing the volume of insulin that you're going to draw out with air. It just helps the process work smoother. If you don't inject air into the air of the vial, and you continually withdraw insulin, you'll create a bit of a vacuum.
This might not be the best visual, but it's all I can think of . . . when we use freezer bags, we put a straw in and suck out the excess air before sealing the bag. The bag is then sucked up against the food that's in the bag. That's a vacuum.
With the insulin, if you keep removing insulin without adding air, you're also going to create a vacuum. Look at the part in the video where I drew up the dose from the pen and it sucked the insulin back out of my syringe and into the pen. That was a vacuum. Think about your vacuum cleaner that sucks air in - it would be like trying to withdraw a dose against that vacuum effect that is trying to suck the dose back in.
Nothing dire will happen to the insulin in your vial if there is a vacuum in the bottle, but it will make it harder to draw up your dose and the risk increases that the dose might get sucked back into the vial. There's nothing wrong with the insulin going back into the vial except that with our modern syringes, there is a lubricant inside the syringe. It's the lubricant that's the problem. The lubricant can contaminate the Lantus. As expensive as it is, most people want it to last as long as possible.
Your vet likely learned how to draw insulin on earlier insulins. I don't know much about them, except the one called "Regular" insulin (Humulin or Novolin R). That's a fast-acting insulin. It's one that I used with Punkin, in addition to Lantus, because he was a high dose kitty. The only point of me telling you this is that R seems to be practically indestructible. I had one vial for punkin that i used for more than a year, maybe even 2 years. It didn't need to be refrigerated and it seemed "tough." It was far cheaper than Lantus too. I just wonder if the technique your vet demonstrated for drawing the dose is perhaps because he learned on insulins that weren't like Lantus.
In any case, I'd assume you want your expensive Lantus to last as long as possible, so I'd treat it carefully.