Earlier quoted context omitted.
Depends if the balls are placed back or not. Similar to cards, any game which never replenishes the deck and lets you draw all the cards down to the last one lets the last pick (but not the prior ones) be non-random.
I literally stated ‘if the balls don’t get put back’. And the last card was still random, btw. But can be guessed now with perfect precision due to the process of elimination. Random at shuffle doesn’t mean unguessable or unpredictable as the game goes on.
I disagree the last card is "still" random, at least in the sense of statistics. It "was" random up until the measure of uncertainty of the next event reaches 0 (i.e. entropy reaches 0). At that point it's no longer a "guess", there is no uncertainty and the remaining pattern is always 100% predictable in that regardless which proceeding events occured to get there it can always be known what the next value is without uncertainty. Since there ceases to be any uncertainty in what the remaining pattern will be there ceases to be randomness in the next value generated. That the card's value was not known at n=0 does not affect whether the n+1 card still is/isn't random when n=51. In another form, that you didn't previously know the value of card n=52 with past information holds no influence whether the value is random or not with new information. Statistical randomness is all about what you know of the future predictability, not about how something came about.
This is also true of events which fall into predictable patterns at any point along the path. E.g. if I had a (relatively useless) hardware random number generator that generated random numbers 0-127 once per second until it generated a 0 at n=17, at which point it ceased being able to pull randomness from it's dead circuits and always produced 0 afterward, the first 17 values were all statistically random at the time of their draw but n=[18,inf) are all now predictable and no longer random from that point on.