Overclaiming was one of the reasons they lost. It convinced Luftwaffe intelligence that the RAF were down to very few operational fighters.
The Jagdwaffe alone claimed nearly 2,000 single engined fighters during the battle, out of a total of around 900 actually lost to all causes, including bomber defensive fire.
LW came with in days of winning the battle in spite of ultra/rader by attacking the airfields and grinding the RAF down.
The Luftwaffe certainly managed to "grind the RAF down", but at no point during the Battle did they do so at a faster rate than the RAF was inflicting on them.
More pilots were being killed than the RAF could hope to replace.
This is true for only 2 weeks of the battle, but the Luftwaffe were losing pilots faster than they could be replaced throughout the battle.
As of 29th June 1940, the 109 force was supposed to have a strength of 1171 pilots, but actualy had 1126 available, and 906 of those were fit for duty.
By the 28th September, established strength had fallen to 1132 pilots, but only 917 were available, and only 676 were fit for duty.
In contrast, RAF pilot strength grew throughout the battle, reaching over 1400 by late September 1940. The RAF has a list of pilots who fought during the battle, and those who were killed. Just over 500 were killed, but around 2800 actually flew at least 1 operational sortie with a fighter squadron during the BoB.
Their replacement pool came within just a few days of being completely empty.
Even if the RAF had no replacements available, and they did, they were still in a better position than the Luftwaffe. RAF pilot casualties (killed wounded and captured) were similar, or less, than Luftwaffe fighter pilot casualties. (The Luftwaffe of course losing far more as prisoners than the RAF).
Now, which do you think is going to break first, the force with 1400 pilots and 1000+ fighters, or the force with less than 900 pilots and planes? Especially considering that the smaller force was training pilots and building planes at a slower rate.
Only when Hitler ordered the LW to shift to trying to bomb London into submission did the RAF turn the situation around.
The Luftwaffe were pressuring the RAF with a very high sortie rate and attacks on airfields. They began those tactics in the last week of August, when they flew nearly 4000 fighter sorties. But they couldn't sustain it, flying only 3200 sorties in the first week of September, then dropping to only 1400 in the second week of September.
That sortie rate should tell you the strain the Luftwaffe were under. To fly 4000 sorties in a week with 675 pilots means each pilot flying 6 times in a week, and much longer sorties than the RAF flew. Little wonder they couldn't sustain it more than 2 weeks.
The British new they were in a fight for their survival and conducted themselves accordingly. Had the LW continued to bomb airfields and aircraft factories England would have been forced to capitulate.
Luftwaffe bombing of aircraft factories was so effective that of the planned 3,602 Spits and Hurris that were supposed to be produced in 1940, 4,283 were actually produced.