This almost veers into what if territory, but I've been rereading Connie Willis's Blackout/All Clear and wondering what historians think of the history in it in general and on one point in particular: how close Britain came to losing.
For those who don't know, the conceit is that time traveling history students from future Oxford are embedded in 1940s Britain but wind up stranded in the Blitz thinking they changed the outcome of the war.
It serves the plot well to emphasize how much the war at this point balanced on small chance events, like a chain that leads from two lost pilots in the fog to Hilter breaking off short of destroying the RAF. Those events happened, and I'm sure everything seemed quite uncertain to folks on the ground, but my hindsight-informed understanding had been that Germany made a mistake even engaging in the Battle of Britain.
(Also, how much does it bother you that she's got future historians doing fieldwork? I can't figure out what they're supposed to be producing from these expeditions aside from good adventure stories.)
It serves the plot well to emphasize how much the war at this point balanced on small chance events, like a chain that leads from two lost pilots in the fog to Hilter breaking off short of destroying the RAF
I've posted about this topic previously. The most important aspect regarding your question is that appearances weren't fully indicative of reality. The British consistently overestimated the strength of the Luftwaffe and thus felt like they were nearing the end of their endurance. The reality was that the Luftwaffe was unable to sustain their losses while RAF strength was replenished and even increased during the Battle. That shouldn't take away from the valor of "the few" but the RAF was far from destroyed. British production of planes and pilots allowed the RAF to savage the Luftwaffe and obviate any realistic chance at a German invasion or capitulation. The aforementioned post goes into more detail on the subject.
I never made it through the first book you're asking about. I started it, but it lost my interest within the first few dozen pages.
Edited to add: I forgot to answer this portion of your question
my hindsight-informed understanding had been that Germany made a mistake even engaging in the Battle of Britain
It wasn't an unreasonable strategy to attack via the air. Germany was extremely unlikely to win a fleet action against the overwhelming power of the Royal Navy and submarine warfare was already underway. To have a chance at an amphibious invasion the RAF had to be destroyed or neutralized. I said above that the British overestimated German strength, but the other side of that coin was that the Germans underestimated RAF strength. Despite their knowledge that the Luftwaffe was suffering heavy losses, the Germans concluded that the RAF was nearly destroyed. All that was needed--they thought--was a knockout blow. So it wasn't entirely unreasonable to start by hitting the RAF at their airfields and then consider switching to bombing cities when they thought that the RAF was almost destroyed. The Germans were wrong, and it cost them dearly both in Luftwaffe losses and in that they then had no ability to knock the UK out of the war in 1940.
As always, followup questions from OP and others are encouraged.