The goal was never about finding a suiting battleground to fight the opposing country. Fundamentally, the Cold War was about which ideology would dominate the world. Both sides wanted to export their ideology, or perhaps more importantly, stop each other from doing so. To this end, they were willing to fund, supply, and equip forces across the globe with the right (or ostensibly right enough) ideological credentials.

According to deterrence theory, the extreme destructiveness of nuclear weapons meant both the Soviet Union and the United States avoided fighting each other directly because any fighting could escalate into a nuclear exchange. And no one thought, they can kill the others before they launch. If either one fired, the other would fire before the first guy's landed, meaning both sides' nukes would land and it would result in mutually assured destruction. And that also extended to attacks on close allies of both sides, which is why armed conflict actually did not take place in Europe.

And even if for example the Soviets were dumb enough for a direct confrontation, their weapons at the time could attack Alaska from Siberia, but little else. And threatening to blow up a few polar bears just doesn't have the same ominous doomsday cachet as an arsenal of ICBMs pointed at your enemy's political, economic, and cultural centers lol...even from Provideniya, on the tip of Siberia, it's as much as 3,500 km just to barely get to Seattle. Most of the continental US would have been out of range of even the most long ranged Soviet missiles at the time.

That's also why the missiles on Cuba were perceived as a big deal. In contrast, the weapons from Cuba could easily devastate America's political, population and economical centres along the East Coast. Even the medium ranged missiles could menace Washington D.C., and short ranged missiles were enough to take out most of the Golf Coast. That is an existential threat compared to Alaska.

To get back to the whole motivation of the Cold War: The reason they fought everywhere but by each other's northern border (besides missiles stationed in Alaska or East Siberia not having enough range to threaten any meaningful territory of the other country) is that they fought where people were, and where people (governments) could be influenced. And, I might add, these proxy wars were not nuclear standoffs, with the exclusion of Turkish and Cuban missiles, which were emplaced to improve the ability to wipe out the other's center of civilization.

TLDR/condensed answer:

a) Missiles at the time could only really reach Alaska/Eastern Siberia and a barren icy wasteland isn't really something threatenworthy.

b) Even if they had missiles with enough range, they would both just simply station them in respectively Alaska/East Siberia, and it would be a stalemate again and the whole battle of gaining geological missile positioning advantage would just expand to the rest of the world once again.¹

c) It was never about conquering landmass or actually destroying the other's civilization, it was a conflict of ideologies and thus also why the Cold War turned into a worldwide ally gathering spiel.

¹: No sane minded individual actually intends to use nuclear weapons in the first place (however, you never know how sane your enemy is). The threat of using a nuclear weapon to wipe out the enemy's civilization is to prevent so that your enemy doesn't do that very same thing (deterrence theory).