I'm going to ignore trying to keep the existing Enterprise and instead go for a reboot using the same actors, as the problems with the original wasn't the acting talent.

There is no Temporal Cold War. Anything sneaky happening in the background is the Romulans, but this doesn't become known until later in the show. Hell, you can keep the creepy communications with the Suliban if you want as it still allows for a Big Bad. You just don't get any good guys helping out Archer.

If we are going to give Archer the Pike treatment, it is going to be that Archer's idea is that his success in life is going to be delivering the Warp 5 program, and maybe even the Warp 7 program in the future. Being Captain of the Enterprise is considered by Archer to be a way to help understand what a ship is going to need in future iterations. His rejecting of the call is going to be that Robinson is going to get the future Archer wanted, but Archer is going to have the future that humanity needs.

Keep shows like P'Jem, as they really help show how humanity is becoming more involved in local affairs. Maybe the Enterprise accidentally ruins some small Romulan plots to destabilize the area.

Include a show here about an earlier Vulcan intervention into a post-warp civilization that went badly, with news of the problems getting to Earth

Make the Vulcan revolution a much longer series, including possibly including the Andorians attempting a post-revolution invasion of Vulcan space when Vulcan is weak to secure an advantage over their weakened foe. Archer can step into this as well, as there may be a second Romulus plot to try to replace the liberated government with something else that will be aligned to merge with Romulus in the future.

Back on Earth, the coup attempt is a way to get Earth to seal itself off and focus on improving the lives of humans. Make the rebuilding of humanity uneven and a work in progress. Some of humanity may not want to spend money galavanting across the stars and instead use the resources to focus on rebuilding Earth. Have Archer see that a post-coup attempt Earth is starting to the way of the Vulcan High Command. He intervenes with the Enterprise, creating the first opportunity to lead Earth politics, even if it is just to get through a crisis.

Have some other crises happen, eventually leading to an Earth-Romulus War, with Archer now having to be on the front lines.

This could probably be done with the current leadership of Star Trek, given how they are writing Discovery and Strange New Worlds.