Maybe a little bit but 1) you would have had to have dragged the thing up to where he was and 2) depending on how far gone he was, it would only provide temporary relief and there is the possiblity of having to physically carry the man off the mountain (this would be no easy feat at 28,000 feet, on Everest's steep summit ridge with 10,000 foot drop-offs on either side of you).
Are such enclosures ever used above Base Camp?
Don't know, as I've never been on a major pay-to-climb expedition like this. However, heavy stuff is usually not carried any further than you have to on Everest. My guess is that a well-provisioned (and expensive, since a porter would do the hauling) pay-to-climb service may have one of these at the high-camp (this would be at the South Col on the route we're discussing here). Nonetheless, the range of services available through the guide-service you are on would vary widely.....look at the costs cited in the article ($6,000 -$35,000) and ask yourself what you would be provided with by a $6,000/person climbing service.