I do not know a space equivalent, but in medicine, treating severe diseases teaches us a lot more about treating their mild cases than the mild cases alone could.
As for space races and political motivations, if you look at China's advancements in space recently, both in terms of launches per year and the pace of their construction capability in LEO, it is clear that we are in a race. On both counts (launch count and construction in LEO, demonstrated by Tiangong SS missions), the US is losing it. There indeed does not seem to be political motivation to do much about it.
And tautologically or not, it is important to recognize the value of how much we could learn from a Mars colony that can't be learned on Earth, in LEO, or on the Moon.
1) Luna first, send stuff to space with mass drivers.
2) Gravity well is lava, mine asteroids (near Earth, then Belt) and build massive space colonies to your liking (O'Neill Cylinders, etc.).
3) Lets build balloon cities in the atmosphere of Venus!
4) Settle (likely after doing nr. 2) the ocean worlds of the kerbal-scale Solar System analog around Jupiter.
But most likely, some mix of those will likely happen, with different organizations having different needs and aims (eq. massive free flying factories and scientific instruments vs self sustained city building).
That seems backwards. I would switch Mars and Earth in this sentence. Doesn't it make more sense to take the easier steps first, which will prepare us for eventually taking bigger steps?
I'm not against Mars exploration. There's no reason we couldn't do both: keep sending probes to Mars, maybe even humans in a decade or two, but make a Lunar base the priority. Then apply the lessons we learn on the Moon to establishing a colony on Mars. And literally use the Moon as a springboard for Mars missions.
I think this is pretty much universally agreed to be the sequence of events. A lot of discussion about this seems to imply we'll just proceed with some Mars colonization mission before we understand the challenges it poses, and before we're certain we can tackle them.
BTW, that JFK quote seems out of place in this context, considering it was made at the height of the space race. The speech was politically motivated, and was meant to inspire a nation to not "lose" the race, while instilling fear in the enemy.