OK - lets take a look at some numbers.
Stokes averages 38.42 with the bat in Test cricket. Stokes averages 31.73 with the ball. We are looking for, in the absence of his bowling, someone to bat at number 8 with figures around 23 with the bat (60% of Stokes average) and his bowling average or better. This meaning you can play a 4 man attack without having a rabbit at 8 and then get shot out once you get through your wicket keeper.
Bowling Average - 32 or lower, Batting Average 20 or higher, 50+ wickets taken, 20 matches played
There are 8 guys on this list (who aren’t Ben Stokes) who played Test match cricket in 2020. Out of 55. That’s 14% of the list in a sport that has been going since the 1870s. With Stokes it’s 16% of the total list. Expand a little and 20 of these players (including Stokes) have played Test matches since 2000. That’s 36% of the list.
I would argue, in fact, that it’s never been easier to find someone at Test level to perform within the broad criteria of what I’ve suggested (hell, England have had 3 within the last decade - Swann, Woakes and Stokes - with already given caveats for Woakes on how he performs outside England with the ball). There’s a lot of reasons for this - covered pitches, better bats, expansion of T20 batting techniques giving even limited players more scoring shots - but the major one is probably that teams have looked at their batting line up and said, “to get picked, you can’t be a mug with the bat” since it’s a waste of scoring potential in your side.
Note that these aren’t even really bowling all rounders (if we defined all rounder as batting average > bowling average). Most of them are just bowlers who are competent with the bat.
I don’t really disagree with your formulation of a side (except inasmuch as when England have been good in my lifetime, they’ve generally had a good all rounder to balance the side, with an exception which I’ll come to) but England’s problem is that they currently can’t bowl 4 bowlers in the side, assuming Stokes only bats. This is primarily for 2 reasons: their spinner isn’t tight enough to bowl 30+ overs with the old ball, to spell their pacemen effectively and Jimmy Anderson is increasingly unable to take wickets outside his opening spell or two of the match (no wickets in the 2nd innings of either the games he bowled in v WI, no 5 wicket haul in a second innings since the back end of 2017).
When we did bowl 4 bowlers and were successful - back in Oz 2010-2011 - we had Swann. And there he is, in the run I linked to above. Precisely the type of player (absent all time great bowlers), who allows you to bowl 4 bowlers and have someone competent bat at 8. That 2010-11 side also had Bresnan in it, who knew which end of the bat was which too, even though he doesn’t qualify for the run above.
In short, I don’t think it too outlandish to suggest that England find one of these players - everyone is finding them and England should be able to too. England need a tighter spinner to bowl in a 4 man attack. England need to build the plan for life after Anderson - they may even consider retiring him.
Also, riffling through the link above, Paul Reiffel was substantially better than I thought. England could use someone with Paul Reiffel’s career now. I knew Ryan Harris, Brett Lee and Mitchell Johnson were all good with the bat but Reiffel’s numbers are damn good, albeit over only 35 matches.