Hiya!
The shortest answer to your question is that the stereotype is highly inaccurate, and one or the many unwelcome relics of the common misconception of the allies in World War One being 'lions led by donkeys.' While Great Britain's generals were numerous in number and varied in personality during WWI, and there are obvious dangers in making sweeping generalisations, there is strong consensus supporting the mediocrity of the BEF's commanders. That is to say, like every other large organisation, Great Britain's generals varied widely from incompetent to brilliant. This same variation should be expected when discussing regard for human life - within reason. A general who callously wastes the lives of his men is a bad general, and would be in the minority.
In the case of Haig, the oft-repeated myth of his disengagement, callousness and incompetence as depicted in Blackadder Goes Forth and numerous polemics of him written throughout the 20th century has been thoroughly debunked.
There have been some fantastic in-depth discussions of the "Lions led by donkeys" myth here on /r/askhistorians. I'm on mobile at the moment, but I'll happily dig them out for you once I'm back online!
I hope this helps!
The idea of Haig's callousness towards losses derives in a large part from the post-war publication of his diary. Most infamously, he wrote that he did not think that the casualties on the First Day of the Somme could be considered to be excessive, which naturally rings rather hollow to our ears. However, it's worth noting that Haig was a notably inarticulate man even among members of his own class: plenty of primary sources from the period before and during the war point to Haig's awkwardness in conversation, so we shouldn't take this as proof that Haig didn't care about losses. Furthermore, an officer does not rise as high as the position of Commander of the British Expeditionary Force without a good dose of moral courage: everyone from Haig to the lowliest Private soldier knew that the BEF would take casualties in the Somme offensive, and Haig could not simply "switch off" the battle after one day of nasty fighting, not while French troops were dying in equally incredible numbers at Verdun further south.
It's also worth noting that Haig worked absolutely heroically after the war to secure proper welfare for his veterans: he refused all honours until he judged that the government had made appropriate financial provision for his soldiers, and delivered a scathing denouncement of the Armed Forces' pension policy to a parliamentary committee in 1919, which was so embarrassing that Lloyd-George's government was forced to change tack. He was an instrumental figure in the coordination of various disparate veterans' groups and the eventual foundation of the British Legion in June 1921. He was seen to work late into the night personally answering correspondence from his veterans, to the point that the BEF's old chief medical officer, Colonel Mickey Ryan, considered it a significant factor in his death from a heart attack in 1928.
Sources:
Gary Sheffield, The Chief: Douglas Haig and the British Army
Gordon Corrigan, Mud, Blood and Poppycock
I think it's worth referring here to John Terraine's excellent essay on the matter which can be found here and from which I'll quote the below as a summary. I'd be happy to elaborate if you want :)
But of course, the cost was high - during that period from July 1916 to November 1918, the British Army in France had over two million casualties (just about the same number as the French in 1914 and 1915, when they were engaging the enemy's main body). In the next war, it was the turn of the Russians to do that, between 1941-45, as I have said, it cost the Soviet Army about 13 million in dead alone.
In both wars, the German Army proved to be very hard to beat. In the Second World War, the British did not see very much of it, which accounts for their much lower casualties; but in the Great War, for the only time in their history, they fought the German main body and beat it, and as I have stated, if that had not cost us appalling casualties as it had the French and as it would the Russians, it would have been a miracle requiring, supernatural explanation. The disciplines of coalition war pressed hard on the British Army between 1914 and 1918: they compelled it into actions for which it was not ready and not equipped in 1915; they forced it into a leading role in 1916, before it was really fit for that; and in 1917 they forced it into strategies in which the British Command did not believe, and which cost the Army dear. All this came as an unwelcome surprise, but in truth that pales into insignificance beside the real burden of novelty which that unfortunate generation of generals had to bear.
[The Somme & Verdun: fruitlessly wasteful, or painfully necessary?] (https://www.reddit.com/r/AskHistorians/comments/32t8bn/world_war_i_question_largescale_offensives_like/)
[Did people REALLY march into machine gun fire in 1914?] (https://www.reddit.com/r/AskHistorians/comments/33fcrk/is_it_true_that_there_were_troops_marching/cqkdhx5?context=3)
[Passchendaele and Haig] (https://www.reddit.com/r/AskHistorians/comments/345l7x/why_did_world_war_i_soldiers_not_revolt_against/cqthml6?context=3)
[Why did WWI Soldiers not revolt or refuse orders to go over the top?] (http://www.reddit.com/r/AskHistorians/comments/345l7x/why_did_world_war_i_soldiers_not_revolt_against/)
[During WWI, what role did Batmen or Valets play in combat?] (http://www.reddit.com/r/AskHistorians/comments/2ycfmr/during_wwi_what_role_did_batman_or_valets_play_in/)
There have been lots of excellent answers here; I'll just post some I've given!
In short, 'Lions Led By Donkeys' is one of he many bankrupt myths of the Great War, a war which saw 47 French, 78 British, and 86 German Generals killed in action.