selling stocks and buying bonds in a recession where interest rates go down (so bonds are priced higher) and expected revenue for most sectors go down is a no brainer.

Agree, but there's also the expectation of future debt. A bigger debt means a bigger issuance of bonds, so the new supply-demand equilibrium moves the rates higher as it's the only way to convince investors to allocate more to bonds.

Technically, yes, but if you create demand for long term bonds and create supply for short term ones, you'll flatten the yield curve and pay long term yields for short term durations.