The 'rumor' is more likely to be from someone useing bolts, nuts, springs and washers (paper/plastic/rubber) and it's been retold as 'screws'
M4 bolts/nuts etc can be had was a quality hardware store for pennies.
You don't 'need' a back plate - but bolts will bend the mobo more that push pins, so a back plate will remove the risk of damage.
You can invent the mounting (simular to how waterblocks are fixed) with the bolt head at the back - this way swapping coolers/block doesn't need the mobo removed.
Edit:
Yes to install bolt through kit involves taking the motherboard off to fit backplate. But tbh if it ain't broke don't fix it.
I've fitted mine with bolt through kit (albeit not thermalright) and the temp went UP by a few degrees. Since then I've even lapped both cpu and heatsink and the temp still wasn't as good as it was.
Did you use a cheap TIM or AS5.
The freezer has MX-2 on them with is one of the best anyway - if you used AS5, the it's has a burn in time (200h I think) in which temps will drop 3-5c
Cheap TIM is well cheap.
Also if your ISH is not level it's very easy to not apply an even presure with a bolt fixing.
Edit2:
removing the push pins is easy with a junior hacksaw / tin snips / shape knife.
But I'd not use those case screws.
If you don't like 70c+ get a tower cooler or drop the OC