The BEF required a lot of time to expand at the outset and then to reconstitute itself after each major offensive. As such the BEF would go through periods of less fighting, which got on the nerves of the French. The French would chide the British about not fighting, and that they were doing all of the work - which is a charge that has some validity up to about 1916. In addition, the British tended to have a much higher ratio of soldier-to-front-footage than the French, which also got on French nerves.
The BEF entered the war with about 160,000 soldiers in France, compared to a French army of about 1.0 million and a German army of about 1.5 million on the Western front (Hart). More specifically, the French army of 1914 was standing at about 884,000 troops that could be expanded to about 1.7 million with immediate reserve call-up, and have another 1.1 million in the recruiting pipeline after that (Mosier).
The BEF had a long way to go in terms of manpower. It started with the 160k soldiers mentioned above. By January, it was 265k soldiers holding 40 km of front, on a Western front totaling 715 km (Mosier). In January of 1916, the British had 1.1 million soldiers on 96 km of front, of a total Western front of 646 km (Mosier).
Naturally, this was a sore point with the French, who were beginning
to wonder why it was that the BEF required nearly as many troops to
hold down a small portion of the front as they did to hold down the
entire front. Mosier, pg 230.
As such, we can see that there was some friction between comrades as to the ratio of soldiers to footage of the front.
In terms of combat action, the French were also bearing a heavy burden.
After the early phase of the war, France had enough soldiers to continue to mount offensive operations on various sections of the front simultaneously. A quote by Jean Bernier in his piece La Piercee where offensives occurred "from Mametz and Carnoy to La Boisselle...Then...there was the Argonne, les Eparges, the bois d'Ailly, the bois Le Pretre..." And it is known that many offensives continued to occur, as it is well established that Joffre was energetic and forceful in applying them.
From casualty figures, in 1914 the French on the Western front are believed to have suffered 304,124 KIA, compared to 17,174 British KIA. For 1915, it is 210,879 French KIA to 66,415 British KIA (Mosier, pg 12). From this morbid calculus, we can see the losses of the early French offensives, and the very likely source of the quote in the OP.
The lack of preparation in the British Army shows up in other areas beyond simple manpower. The British were behind in artillery and shell production - two factors that were of vital importance in that war. According to Hart, in June of 1915, the British had only 71 heavy guns to 1,406 field guns (field guns being antiquated by this war, as indirect fire had come of age), and British shell production was a paltry 22k per day compared to French production of 100k and German-Austrian production of 250k.
As for Sir John French's role in this, it is very likely that he in fact saved the BEF in its early days. He was aware that in France he had the lion's share of his country's army, that it was a drop in the bucket compared to the quantities of French and German armies around him, and at Mons and Le Cateau quickly found that the Germans had enough firepower to destroy his army if given the chance. He therefore retreated as rapidly as possible to preserve the army, and talked about getting out of France altogether (from Young and Lawford's History of the British Army related through Mosier).
Further along in the war, politics would plague Sir John French's attempts to get the BEF up to strength and fighting on and for advantageous ground, while British and French political pressure would force him to squander his troops on dubious operations on very poor ground, and the litany of these debates and of French's sometimes dubious compromises is well documented in Hart.
Liddell Hart, B.H. A History of the First World War. PAN, 2014.
Mosier, John. The Myth of the Great War. HarperCollins, 2001.