Here is how I understand the situation.
Let's assume a booster pack has 3 cards. 2 are non-holo, and 1 can be either a regular rare or a rainbow rare.
I have 3 boxes of cards.
Box 1 - has 18 non-holo rares.
Box 2 - has 6 regular rares.
Box 3 - has 3 rainbow rares.
I have created a machine automatically makes packs by taking 2 cards from Box 1, and 1 card from either Box 2 or Box 3. I want a very generous pull rate on Rainbow Rares, so I've programmed it so that for every third pack, it takes a card from Box 3 instead of Box 2. If there are not enough cards from either Box 2 or Box 3, it stops making packs, an alarm goes off, etc.
Pack - [contents from Box 1, contents from Box 2, contents from Box 3]
Pack 1 - [2,1,0]
Pack 2 - [2,1,0]
Pack 3 - [2,0,1] – 2 cards remain in Box 3
Pack 4 - [2,1,0]
Pack 5 - [2,1,0]
Pack 6 - [2,0,1] – 1 card remains in Box 3
Pack 7 - [2,1,0]
Pack 8 - [2,1,0]
Pack 9 - [2,0,1] – 0 cards remain in Box 3
This is my complete run. I have 9 packs, with a 33.333% rate of including a rainbow rare.
Now, suppose someone stole a card from Box 3;
Pack 1 - [2,1,0]
Pack 2 - [2,1,0]
Pack 3 - [2,0,1] – 1 cards remains in Box 3
Pack 4 - [2,1,0]
Pack 5 - [2,1,0]
Pack 6 - [2,0,1] – 0 cards remain in Box 3
Pack 7 - [2,1,0]
Pack 8 - [2,1,0]
Pack 9 - [2,0,0] – Insufficient cards in Box 3. Rather than seal the Pack, an alarm goes off.
I think people might be assuming, is that Pack 9 was being filled with a card from Box 2 rather than Box 3. In favor of the printers, I'm going to assume that that wasn't the case. So, no problem, right? Well, no, because you'd also need to hold back Packs 7 and 8, too. As it currently stands, for Packs 1-8, you have 2 out of 8 packs with a Rainbow Rare – great on you for not releasing Pack 9, but you've still ended up reducing the odds of pulling a RR from 33% to 25% – unless you held back Packs 7 and 8 as well!!!
I'm making a lot of assumptions here, but in most thought experiments I can think of in terms of trying to understand how these thefts could've not affected pull rates, and in many cases, the only way you can make that claim is by necessarily separating even those potentially affected packs. There are other scenarios where it isn't necessary to separate packs (i.e. if the response to an alarm on Pack 9 is to refill Box 3, then there is no problem whatsoever) but it really depends on what their defined process was/is for handling a shortage.