Perhaps Kremlingames/Nostalgames might be of interest? Their first games was titled Crisis in the Kremlin. It allows you to guide the soviet union and create and live through alternative realities (what if the USSR had turned trotskyist?, what if OGAS succeeded?, what if the soviet union hadn't left Afghanistan?, what if things escalated to direct armed/nuked confrontation, etc). It is however quite an unintuitive game and you would probably need a guide to play it. They are currently working on a sequel, though (Crisis in the Kremlin: The Cold War). However, they have newer games. One of them being China: Mao's Legacy, which I think with the Fallen Eagle DLC, based on the reviews, allows you among other things lead to the collapse of NATO, but I haven't played that game yet.

Its a boardgame so you could play it on boardgame sim probably, but Twilight Struggle is pretty fun when you play as the USSR. The Soviets are really strong in the early game and can steamroll the Americans

I think the closest you can get is the Cold War mod for Victoria 2

I recently attempted to try out Heart's of Iron TNO, after it's most recent update. And if you can look past the eye burning UI, It's got a fairly okay storyline with a firmly anti-fascist narrative. Although if you're specifically looking to trash on the US, then for strategy games I could recommend Eugen system's Wargame and Warno series. I remember playing one of them way back in the day, and it was pretty good. They released the latest entry this year, I haven't played it but it seems to have good reviews.

Its a little janky and made by a Frenchman but this free browser game is low key the best in this genre also includes PR China and france (you can go communist):

https://cold-war2.com/en/index.php

there's a sequel in the works and the dev makes lots of interesting historical games. Check out out his FBI simulator too.

Unfortunately there isn't. The only decent one I know is the 2003 RTS Rise of Nations, which has a mode that allows you to play a very limited form of grand strategy set during the cold war (you play some cards, take a couple actions, then move your armies to enter into an RTS match with the military of another country)

I've been looking for years for the same kind of game, but it looks like nobody is interested in making it, or everyone thinks it's too hard to implement. There's a couple games that advertise themselves as Cold War simulators, but they usually only focus on a nuclear exchange, or they are really just a click-intensive RTS game happening on the world map (e.g. Supreme Ruler).

There's an upcoming grand strategy game called Espiocracy, set to release sometime in 2025. I hope it's the Cold War grand strategy we've been looking for.

Until then, HoI4 mods are all we get. The good news is you can mod them further if you wish. Changing words in events is not that hard. All you need to do is find the event in the english localization file and change the offending words. It's just a bit time-consuming if you want to do a bunch of changes.

Seconding Hoi4 TNO. Next option would DEFCON. Cheap, neat strategy game.