The old Sandy/Ivy/Haswell i5/i7 CPUs are still actually great for day to day, as long as the frequency is high enough, but they're definately reaching thier end point for anything more than lower spec or esport gaming.
I mean, for balance I wouldn't partner more than something like a 1650, 1660 tops with a Sandy/Ivy/Haswell era CPU these days, i5 or i7, the IPC and clock speeds are just not good enough (~3.5GHz stock)
For a 4070, you really want to be going modern i5/i7 or AM5.
If you can get a cheap second hand AM4 bundle, then 5800x3D or 5700X3D is also pretty damn good, but you're definately losing substantially on your card, and that seems a shame, when the card will have cost you so much.
Saw a bump going from an i7-10700K to an overclocked 12400 (which performs for gaming ~12600/12700 region or so), with my 3080, so expect an even more noticeable upgrade for you, given the 10700 was higher IPC and clock speed than your Haswell.
It's more noticeable at high refresh rate as well, as you really do need better CPU performance to hold average/minimums at a higher average spot, a drop from 120 to 45FPS, even for a second or two, is much more noticeable than a drop from 60 to 45FPS.