Ryan.. sorry, but to hear about this. We had similar issue in our basement a year or so ago. Some good advice in here, and as usual, some advice from people who have never experienced it and are wrong.
Carpet will dry out and is made of plastic fibers typically, so not a huge problem but YMMV. However, if you have a pad underneath, it’s almost impossible to get that dried out because of the open cell foam construction.
Doesn’t look like it got into your walls, so you should be ok here... but if it did, you can pull your base trim and drill 1” holes behind it that won’t show after you replace the trim. This will allow you to make sure insulation isn’t wet (if it is, you’ll probably have to pull drywall for bottom 10” and get wet insulation out of there). Our walls were dry, so no drywall pulled, just trim and a few holes.
All of this was performed by a local Servicemaster company (at an exorbinant price I must add) and we only had to pay our insurance deductible.
They did use 3 large commercial dehumidifiers for about 8 days, but our basement was much larger than what I saw in your video.
Pad was replaced with new... original carpet stretched, and trim replaced. Good as new and you’d never know it happened.