The fact that Kirchoff's Current Rule is valid means that whatever current flows in flows out. But this will only be valid for a steady current circuit, that is, when there is no accumulation of charge. But there is no reason why there shouldn't be even small accumalation of charge and therefore inconsistencies in current, is there? Somehow it all just doesn't seem very intuitive. Am I looking at something the wrong way? Is there a proper explanation?
We are normally applying Kirchoff's Current Rule to a steady state situation. If there were current loss in the steady state that means charge must be piling up somewhere in principle to an infinite value, which obviously doesn't make sense.
It's certainly true that on short timescales charge can be stored, for example if there is a capacitor in the circuit. But in the steady state the charge stored in the capacitor will be constant, so even in this case Kirchoff's law still applies.
Yes you are right. We have made an assumption that (charge flown in) = (charge flown out). There will surely be an accumulation of charges at least in some part of elements.In the case of accumulation of charges we can find the other parameters like Voltage drop etc., using Maxwell's equations. But by making this small assumption we have greatly reduced all those Integrals and differentials in Maxwell's equation to a simple Algebra of OHM's law and KIRCHOFF's laws. This is what we call LUMPED MATTER DISCIPLINE.