Just being a little pedantic -- kWh (kilowatt-hour) is a unit of energy. kW (kilowatt) is a unit of power, i.e., the rate of energy flow.Is such severe fluctuation normal? Throughout the duration of my charge, it kept rising and falling between 30-80kwh. For reference, I started charging with 39% battery and disconnected at 81%.
I've often noticed varying supercharger power levels. I can think of several possible reasons aside from the obvious one (contention from the paired spot). One of the most plausible is site demand management; particularly during peak hours, utility demand charges are substantial at commercial sites (that's how they make their money) so it's quite possible that a Tesla supercharger site tries to limit the total demand of all the chargers at a site to stay under some limit unless it's clear that the site can consistently make use of higher power levels for an entire month.
Also, many sites now have stationary battery plants to help limit utility demand, and it's entirely possible that the SC power levels are adjusted depending on their state of charge. Some have PV panels (e.g., Kettleman City) and their production could also be a factor.
If I were Tesla, I would definitely be negotiating with electric utilities to let the utility remotely control a SC site's total demand in real time in exchange for substantial rate discounts. The entire electricity industry is definitely headed in this direction, and EV charging is one place where it makes a lot of obvious sense.