It's probably better not to think of this as a randomized algorithm. Here is a simpler example of what I mean.

Suppose you have two urns in front of you. One urn is full of N white marbles, and the other urn is empty. Your task is to take a marble out of the first urn, paint it either red or blue, place it in the second urn, and then repeat this process until the first urn is empty. Moreover, when you are done, something very close to two-thirds of the marbles in the second urn must be red.

The catch, of course, is that you have very poor short-term memory, so you can never remember how many marbles you've painted or what colors you've painted them.

The "randomized algorithm" solution would be to use a pseudo-random number generator to produce a number x between 0 and 1 for each marble, and to paint that marble red if and only if x < 2/3.

But there is a non-random way to think of that procedure. Suppose instead that, before you start painting your N marbles, you set out a box of N poker chips, of which you know (that is, have reason to be highly confident) that very nearly two-thirds are red. You then proceed to paint marbles according to the following algorithm. After taking a marble in hand, you select a poker chip non-randomly from the box, and then paint the marble the same color as that poker chip.

This is a non-random algorithm that you can use with confidence, but which requires no memory. And, as I see it, the method with the pseudo-random number generator amounts to the same thing. By deciding to use the generator, you are determining N numbers: the next N numbers that the generator will produce. Moreover, if you know how the generator is constructed, you know (that is, have reason to be highly confident) that very nearly two-thirds of those numbers will be less than 2/3. To my mind, this is functionally identical to the poker chip procedure.