How correct is the statement "lions led by donkeys" ?

As the phrase got most of its force from Alan Clark's The Donkeys, you might be interested to read a revisiting of that work in a review here and in a related article posted by the British Library here. Clark's great fun to read, even if guided more by fury than fact.

There's also an interesting project and series of articles posted by the University of Birmingham to assess the lower "donkeys"; the general officers below Haig, Plummer & Co.: who each really was, and what they did or tried to do. Browsing, you can see quite a lot of variety of aptitude, flexibility, and ability: the contrast between the ineffectual leadership of Thomas Pilcher and independent thinking of Philip Rynd Robinson, for example.

But among all the many complaints about Clark, though I am not a WWI specialist I do think he could be slammed more for not making more of the role that simple bad luck plays in any chaotic enterprise. A great example of this is poor Spencer Scrase Dickins, whose article in the Birmingham project ends

There was no doubting Scrase Dickins’s personal courage, which was exemplary, or his devotion to duty. He was a regular visitor to the front line and his spartan routine included an open air bath in the square at Montauban, even under shell fire. He had the misfortune to command a brigade in an army that had been de-skilled at every level by its rapid expansion and to lead it in offensive operations against a skilful, resolute enemy, entrenched on the high ground. Whether he had the tactical imagination or the powers of leadership to succeed at the level of command to which the outbreak of the Great War unexpectedly raised him is difficult to tell.

In other words, neither a lion or a donkey could overcome a very bad hand, if it was dealt them.