It comes down to the fundamental dillema of Test selection, and why you're trying to place an all-rounder in there in the first place. Ideally a Test side of 11, will need 4-5 bowlers, and want to bat down to 8 or 9. A bowling all-rounder who is of front-line bowler quality directly addresses that dillema, wheras a batting all-rounder, even of specialist bat quality will not. I mean that's cool if an all-rounder could bat 6 and up, but if he's only coming in as bowling option 5, well that's something that can be accomplished by one or more part-timers among your specialist bats anyway, with not too much value lost. The all-rounder only really needs to bat at 7 or even 8, if you have a decent batting WK ( which in more recent modern Test cricket, let's face it you do, as specialist keepers aren't really a thing).
Think of it this way, if you could get 3 bowling all-rounders of roughly specialist quality vs 3 batting all-rounders of roughly specialist quality, which are you taking? Well, with the latter you've really gotten almost nowhere in building your attack. I guess you could pick 3 specialist bowlers very comfortably, and the all-rounders do the remaining bowling job in aggregate. Eh, that's not that much "extra" value. With 3 specialist quality bowling all-rounders you've almost rounded out your attack, and are guaranteed to bat deeper than most teams. Just tack on an "ace" specialist bowler and the attack is done ( 4 bowling all-rounders would be overkill, as you'd always want the extra bowling quality over batting at 11 or even 10 most likely ).
You've phrased that argument in such a disingenuous way that none of it can be taken credibly.
First of all from your previous post, Sobers was a front line batsman and also a front line bowler of test quality and was either the 3rd or 4th option and often bowled the most overs in matches.
I will also say that Imran and Hadlee were legitimate top 10 ATG bowlers who could also bat, so yes there are bowlers who can more than fill the bowler all rounder role as well.
But if I follow your argument, and to quote you bowlers of "roughly specialist quality" who can bat would be better than having 3 specialist bowlers. So we're leaving the ATG realm and looking at fundamentals.
Why would I jeopardize the strength of my bowling attack, when it's the primary role, to add to the batting when you have 5 or 6 batsmen for whom that's their primary job. That's just idiotic, so contrary to your argument, it would be detrimental to the overall quality of the team. If you want to argue for one person at 8, then fine, but not down to 9 or 10.
And to your strawman argument that you would prefer 3 bowling AR to 3 batting AR, you don't need more than one batting AR, and that is also a role that has some importance to be able to be at the very minimum, assist with the rotation give the front line guys a rest without releasing any pressure or advantage gained. And if you do have to sacrifice for someone "roughly of specialist quality" it's one rather than 2 or 3, and even that would be doubtful.
I'm not downplaying the importance of having bowlers who are handy with the bat, but not trying to diminish the alternative either, nor for my "personal fetish" of having batsmen who are slip specialists either.
All are valuable and balance is important to any team, it's valuable to have a good fifth bowler, two or three good slip catchers and any runs the tail can produce are always useful, especially in support of a top order bat, but specialists, especially if they are clearly better than the alternatives, do have an important role. Because at the end of the day you have to choose the batmen most capable of scoring runs and the bowlers best suited to bowl out the opposition for the least amount of runs scored.
