I shall do a proof mathematically (well, semi-mathematical) first, then comment on why the given method doesn't work. Also, just to be clear, what the student did was use intuition. If he/she actually worked out the probability he/she should get the correct answer as well (I think the error in the argument of the thought experiment lies in that the extra marbles in Jar B was not taken into account)
Let handful of marbles = x marbles.
By 'new marbles' I mean 'non-original marbles'
When you take out the marbles from Jar A
Jar A has: 100 - x red marbles
Jar B has: 100 blue marbles, x red marbles = 100+x marbles.
Now you take marbles out from Jar B and put in Jar A.
Say m red marbles and n blue marbles
Jar A has: (100-x) red marbles + m red marbles +n blue marbles
So Jar A has: (100-x+m) original marbles and n new marbles
Jar B has: 100 blue marbles + x red marbles - m red marbles - n blue marbles
So Jar B has: (100-n) original marbles and (x-m) red marbles
But since the number of marbles taken out is the same, x = m+n
Original marbles in A: (100-x+m) = (100-n) original marbles
New marbles in A: n = (x-m) new marbles
which corresponds to the expressions for the marbles in B.
So it's clear that the number of original marbles left in each jar is independent of the proportion of red and blue marbles you take from Jar B.
Why? Because the more red marbles you take out of Jar B, the less number of new marbles there will be for both Jars. i.e. the more red marbles you take from B, the less number of new marbles are left for B. That also means less blue marbles are transferred to jar A, leading to a decrease of new marbles in that jar as well.