I've never seen him do anything like that at Burnley and that's where he made his name. I don't follow Burnley extensively and have seen them a few times they played us and they are well known for playing low block attrition football(even against teams like them) where there is very limited need for sweeping and Burnley are well known for hoofing pretty much everything upfield.
Yes he may be good at claiming crosses but we haven't faced a serious problem of conceding corners or De Gea being constantly targeted at set pieces.
De Gea is a far better shot stopper than all goalkeepers other than Allison/Ederson. I have seen De Gea making countless saves that the likes of Pope and Henderson can only dream of. So unless we find a goalkeeper who is an excellent sweeper and has good shot stopping abilities, we have to stick with De Gea. No point in wasting money/time in options like Pope.
As sullydnl pointed out above, that really isn't the case. Pope and De Gea are both at the extremes in terms of defensive actions outside the box, with Pope doing it more than just about anyone while De Gea does it less than just about anyone.
These are their stats compared to all keepers in the top five leagues over the last 365 days:
Nick Pope
De Gea
Pope even comes out more often than Alisson (1.63 times per game - 97th percentile) and Ederson (0.85 times per game - 71st percentile) and comes out a greater distance (18m and 17.1m respectively). Now obviously those two are in teams who dominate the ball so there's probably less chances for them to come out, but Pope is eager to rush out whenever needed. Like I said it's just the quality of what he actually does with the ball that holds him back.
Your point about De Gea's shot-stopping ability was true up until 2018. Since then he has been very very average other than a few months at the beginning of last season. Sure he'll still pull off the occasional world class save, but for every one of them he makes he also makes a mistake and lets in what should have been an easy save. To put it in context, over the last four full seasons he's had +0.00, +0.06, +0.00 and +0.18 in terms of how many goals he saved per 90minutes compared to what the average keeper would be expected to. Meanwhile Pope has had +0.29, -0.05, +0.15 and +0.12 in those same four seasons. So one season where he did disappoint, but otherwise comfortably better than De Gea even in shot-stopping quality.
The only question mark over Pope is whether he'd be able to handle the pressure of playing for us. If he handled that fine then I have no doubt he'd be comfortably better for us than De Gea is. Still not what we'd ultimately want, but a definite improvement in the meantime. Of course now that he's at Newcastle that's off the table; the time to get him would have been before this season started when he was available for cheap.