Learnt something new! I could line the inside of the entire cupboard with this type of material
Beware insualting the cupboard as a side effect. You don't want the pc overheating. I'm surprised yo have it in a cupboard unless it's open backed as poor airflow could cause all sorts of issues.
If you use a sound insulation mat underneath, it should reduce the transmission of the vibration to the cupboard as the only wy for vibration to be transmitted is by direct contact with a surface so only your base via the feet is in contact and transmitting the sound. What will then happen is the vibration becomes amplified by the cupboard which actas as a "sounding board" ie speaker. Insulating the whole cupboard may reduce transmission by the cupbaord to the air, however, as I said, when you insulate you also insualte for heat so could increase temperatures inside. The 1st step in my opinion is reduce transmission from the PC to the cupboard through the feet. If that doesn't work you could consider other measures but beware the heat issue.
Also don't overlook manula dfragging. If defragging is the causes, simply turning it off and doing it at a time when it won't disturb you is a very simple answer. I don't think you can automatically specify a time, so it would be on you to remember, but it's not as if the world will end if you forget 1 day or are a few hours late. Easiest way is pick a"safe time" and do it the same day / time each week.