I'll work through one session to try and explain my concern.
At qualifying pace on the softer end of the tyres available usually there's 1 hot lap of outright pace in a set, maybe 2 at some tracks.
Q3 is 16 minutes, with the first elimination at 7 minutes in.
That means that all 22 cars will have to have set a time within that first 7 minutes, so that will be using that tyres 1 lap of outright pace.
From then on there are 2 options:
If theres no pit stops - you have 9 more minutes of drivers on ever degrading tyres going slower, rendering that 9 minutes utterly pointless.
If there are pit stops - nobody who is within the next 2 or 3 places to be eliminated can make a stop and get out in time to try and post an improved time, so will be eliminated anyway. Rendering most of the remaining 9 minutes utterly pointless.
In summary, I can see a system where the best times in a session are all posted within the first 7/6/5 minutes, and the remaining 9 minutes of the session are just people unable to improve waiting around to be eliminated.
I also don't think that dwindling the field down to 2 cars at the end is exciting. There were complaints when the rules mean lots of times we only saw 6 or 8 cars in Q3, under these rules you would have only 8 at most, and the number would drop to just 2 at the end. Currently (ignoring Mercedes dominance) with 3 minutes to go in Q3 we could theoretically have any one of 10 people on pole. Under these proposals, with 3 minutes to go we will know it will only be between 2 drivers, which could be a) on old tyres not likely to improve, b) in the pits or making a pit stop meaning they won't have time to improve, or c) out on track and going for it, but only 2 cars.
A 10 way shootout for pole is more exciting than a 2 way one.
And then theres the Red Flag issue, which basically kills the whole idea on its own for me.