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I think an overlooked factor not mentioned yet is they are tired.
For the last 4 to 8 years you lived with the knowledge that there was always a chance you'd get woken up at 2 am, or pulled out of lunch, or whatever and told that the military had detected what it thinks is an incoming Russian full scale nuclear strike, and you have 3 minutes to decide if you want to kill 100 million people in response to this suspicion. 
Like what the fuck. Sure. I can type those words. But the situation is so insane I'm not going to pretend to understand the emotions and stress that must come out of being actually in that position where that phone call might come to me. 
And that's just 1 tiny part of a president's life. You then take into consideration that they are basically working a shift for 4 to 8 year straight. You never really are off work when you're the president. The decisions you make daily are life and death. The "lesser" decisions you make might create a million poor people instead of lifting a million out of poverty.
The meme of the presidents aging 10 years in a 4 year term is a meme for a reason. 
If I had just gone through that, I'd want to be done working. I'd have literally 0 interest in going back on the campaign trail to get a job in the house of representatives. Those 4 to 8 years were more than enough given to my country. I'll take whatever left in my life back for myself.
For retired presidents they usually feel they can use their platform to focus on projects that are important to them, and only dip back into the world stage if they feel they can bring weight or influence to something. Carter and Clinton have both negotiated prisoner releases with North Korea, for example. Former presidents have also been influential in raising funds for humanitarian relief (Clinton and Bush Sr worked together after the tsunami in 2004). Also don’t forget that ex-presidents can make an absolute fortune for speaking engagements, and also on book deals. But really, for most people that serve as president, that was the culmination of a lifelong goal. They are ready to move on to something else. Eisenhower painted and golfed. Teddy Roosevelt hunted and wrote books. Jimmy Carter is a Sunday school teacher and builds houses.
Other statesmen do occasionally get back into politics, and John Kerry is currently the special envoy on climate. After Mitt Romney lost his presidential election bid in 2012 he moved to Utah and was elected to the Senate and is still there. 
There is precedence (though it was a long time ago) for former presidents to keep serving in other offices as you suggested. John Quincy Adams was elected to the House of Representatives after he left the presidency. William Howard Taft was appointed Chief Justice of the Supreme Court, and by most accounts he was much better suited to that role than the presidency. >

Teddy Roosevelt ran against his successor (the candidate he earlier had supported/promoted). He is an exception to the rule, though. >

And after he lost that campaign he went to South America and almost died mapping a river in the Amazon. His entire life was an exception to the rule. >

Teddy Roosevelt was an absolute badass. >

Then proceeded to speak for something like an hour and a half. He’s basically the Chuck Norris of US politics. >

Considering Chuck Norris was an actor who only played someone tough on screen, I'd say Ronald Reagan is the Chuck Norris of US politics. >

He was world karate champion, so a bit more than acting tough