and the strong recovery the UK has had
"Strong recovery". It took until 2015 for the UK to exceed the GDP/capita it had in 2007, 8 years earlier. It's still miles from any reasonable plotting of "trend". About the only thing good that can be said of the UK's recovery is that growth is now on trend, but seeing as it was flatlined for so long it needs to be well and truly above trend to ever hope to recover.
There's just no reasonable interpretation of the figures where the UK has had a good, let alone strong, recovery. Taking 8 years to get back to where you started is not good.
There's more to monetary policy than just interest rates.
They all amount to the same thing. QE is about tweaking the yield curve. UK doesn't have reserve requirements, so I don't know what you're going on about there, but tweaking that is just another way of tweaking interest rates for a given level of reserves ultimately. Adding/subtracting reserves does the same thing in a much cleaner fashion (and paying interest on reserves and offering discount window lending is cleaner again).
Interest rates are low because the BoE is targeting inflation, not because of general economic growth.
Yes, and it's hard to have inflation when the economy is starved for demand.
Any textbook will tell you that a government sucking demand out of the economy will be met by a central bank trying to boost it through lowering interest rates. Austerity = lower interest rates always. (assuming you have your own free-floating currency and inflation targeting central bank)
But the government isn't the sole actor in markets, and government withdrawal can boost the private sector as it's no longer crowding it out,
Crowding out? Limited loanable funds? Please, what era are you from.
The GBP has been floating since the 90s. It's not in limited supply.
I mean, sorry if that came across as harsh but it's just a ridiculous proposition that the UK gov't taxing people less / paying people more somehow "uses up" pounds.
It did, back in the day, when pounds were just a wrapper around gold and/or USD and/or Deutsche Marks (it's been all of those at various points in the past century). Back in those long past days you'd "use up" GBP by borrowing it. There'd be less gold to go around, pushes rates up, private sector can't borrow as much (because interest rates are higher!). Under those old antiquated models central bank targeted a fixed exchange rate meaning interest rates had to float.
Now the central bank targets fixed interest rates, to pursue goals of achieving inflation considered conducive with full employment, and the exchange rate floats instead. In this model there can be no financial crowding out. Any textbook that tells you otherwise has not been updated since the 90's when the GBP was finally allowed to float. Old knowledge is a dangerous thing, and it takes a long time for these myths to die unfortunately. But you must let them rest.
Today, the government taxing people of less GBP leaves more GBP in the hands of people with which they can spend/invest/consume. It does not deplete it in any conceivable way. And austerity can only be viewed as a higher level of taxation relative to expenditure whichever way you want to spin it. Taxing people more relative to how much you're paying them => contractionary => less money to spend/invest/consume. It's why it tanks growth, exhibit Greece/Italy/etc etc. The UK though, as a monetary sovereign (unlike those nations), needed never have entertained it. It could have pursued simple growth orientated policy but instead it chose the opposite.