Vehicles aren’t generally pushed in that universe even when broken down (they’re space ships or floating cars, why push them?), Princess Leia is referring to another practice, which is known to the characters but obscure to the viewer
Vehicles are pushed when broken down in Star Wars, in the same way as someone on planet earth (our planet) may push a Vauxhall Nova to get it moving if the usual ignition procedure proves ineffective. Princess Leia is sarcastically offering to do this, and Han Solo is fighting sarcasm with sarcasm, developing a crucial character dynamic
Would imagine pushing a broken landspeeder back home would be a pile of suck, given that you would assume the repulsor drive would be shot and it would just be a dumb sled. But hey, you could try it.
In the sense that it has a defining point where it got crap and therefore I no longer believe it to exist? Yes. Yes it is.
(That point, by the way, is when they decided that all of the expanded universe books were no longer canon and therefore attempted to erase my childhood. Well suck it, George Lucas, because for me there are three films and a bunch of fair to pretty good novels and that’s all I need.)
I don’t want to live in a world in which Grand Admiral Thrawn never existed, basically