corned-beef cure/brine: viscous, snot-like substance - is this safe?
Unfortunately I dont have a concrete answer for you but both the gassing and the snot would make me lose confidence that it was safe to eat. However, in this question they person answering said it might be sugars breaking down. Another commenter also lists spoilage signs for corned beef brining:


White slime and foam on top.

Milky color with or without foul odor. Cloudy is ok, milky is not. If it’s milky but has no odor, it’s still bad. The odor will come if given more time.

Brine turns blue in color or has a bluish hue/tint.

Brine becomes very tacky (gluey) to touch. There is a difference between being gluey and being slimy. Slimy is ok, gluey is not.


There are certainly other bad signs, but in general, these are far and away the ones most encountered.

Another commenter here says the slime is normal.
I've seen lengths of brine time span from 5 to 10 days, but if you're worried you could cut it down to 7 days in the future. The USDA seems to be in agreement about the 5-7 days in the fridge at 40F or less.
In summary, I would be a little nervous with your current situation but in the future, maybe a little clear slime after 5-7 days would be fine. 10 days seems to be stretching it, and it seems like you've got a lot of slime in your video.