There are multiple reasons it might stumble after takeoff. Carb ice is only one, though it's certainly a common one.
Ice tends to form faster with the throttle closed or only open a little. That's why the carb ice risk charts show icing worse at low power or on approach. The throttle plate chokes off the airflow, causing the engine to suck hard getting the air around that plate, and the velocity of the air goes way up. Increasing the velocity of an airstream lowers its pressure, and decreasing pressure lowers its temperature, so moisture (water vapor) will respond to that by freezing to the plate and the carb's walls.
So when you're taxiing around on the ground in an environment conducive to icing, ice can form. The engine's RPM will drop as the ice starts blocking the airflow around the throttle plate, so most pilots just open the throttle without realizing what's happening. This is the big mistake, and it's entirely due to poor instruction or simply intellectual laziness on the part of the student. Doesn't want to learn a bit of physics. There are many people who blame "the antiquated designs of our engines" but the fact remains that the affordable airplane will usually have a carburetor and if you want to reach old age you'd better understand it and learn to manage it. Flying is like that: reality doesn't care about a pilot's uninformed opinions.
So anyway, if you accumulate some ice while taxiing and then take off, the massive airflow going past the wide-open throttle plate isn't experiencing the same temperature drop it was at or near idle, and if the temps are right it starts melting that ice. The water and chunks of ice going into the cylinders don't burn very well at all and the engine barfs a bit. That same roughness convinces far too many pilots to turn the carb heat off after pulling it when experiencing power loss, and they end up with a dead engine. Again, poor training and understanding. It's sad. It wasn't like that when I was young; pilots easily learned to manage the engine because many of them still drove cars that had chokes and could ice up sometimes and those engines needed understanding. Now we have cars that largely operate themselves and newer pilots want airplanes that take that load off their minds. Ain't gonna see that anytime soon, so learning stuff the right way the first time is important.
Does this count as a rant?