I'll bring up two points, that I haven't seen before.
I'm not particulary happy with the ruling and think it is both not totally consistent with parts of the rest of the templating and rules in general (though i'll admit i don't think there is a solution the card as written can be ruled to be) and unintuitive. (An admittedly subjective sentiment). I also disagree that FFG’s (and Richard Garfields) interpretation is the only possible way this could be ruled/read.
While never explicitly stated, the "you" in Biomatrix Backup can obviously only refer to its controller, because a lot (?) of other effects use you as well and there has be no one (at least no one i know) who has seriously suggested otherwise.
In example, does anyone truly believe Mother to allow the active player to draw an additional card at the end of his/her turn?
If not why would the same word refer to a (at least in some states of the game) different player on this card and this card only? I don't see any logic reasoning behind that. The notion that the "you" part could pretty randomly and unspecificly refer to different players strikes me as quite frankly absurd.
Does the "If you do" in same cards text probably refer to different players as well "depending on gamestate"? What does that even mean? In which gamestate is which "you" which player? If so how do you know? How is it decided who this "you" is? Do you roll the dice to decide? Does the active player get to decide? Do we the community vote on it? (I'll stop before i get to hyperbolic here, but i think i made it obvious that a text refering to a specific thing/word cannot and should not mean different things based on totally undefined criteria. The very notion strikes me as absurd.)
On the matter of DESTROYED effects the Rulebook tells us this (emphasis mine):
When a card is destroyed, it is placed in its owner’s discard pile. If a card has a “Destroyed:” ability, the effect resolves automatically when the card is destroyed, immediately before it leaves play.
Ruling that this only applies when the card itself leaves no choice makes this whole paragraph either
a) (at least partially) untrue (The effect does not actually resolve automatically in this instance)
b) Completely superfluous as every time it's relevant the text on the card would lead to the same result anyway.
Since i (at least hope) we can agree that it is (or at least should be) neither, this applies here as well.
Just as a quick reminder, the ACTIVE PLAYER section of the glossary (which is largely responsible for this ruling) reads like this:
The active player is the player taking the current turn. The active player makes all necessary decisions for all card abilities or timing conflicts that need to resolve during their turn.
That out of the way, in the case of Biomatrix Backup we now have to decide whether we can resolve the card by these rules or (when this is not possible) to invoke THE GOLDEN RULE.
If the text of a card directly contradicts the text of the rules, the text of the card takes precedence.
To be honest i don't exactly know what actually constitutes a "direct contradiction". At least in other FFG's games it's basically "As long as cou can follow both the rules and what is written on the card, you have to do so".
Whether Biomatrix Backup text is actually in "direct contradiction" with DESTROYED is and ACTIVE player (imo) up to one's interpretation to what constitutes "in direct contradiction".
This leads to three (in my opinion pretty equally arguable) solutions:
a) Don't invoke THE GOLDEN RULE and resolve in accordance with both ACTIVE PLAYER and DESTROYED rules (If you think Biometric Matrix is not in "direct contradiction" with both ACTIVE PLAYER and DESTROYED):
Since according to DESTROYED we have to resolve the effect automatically, there actually is no choice and (the controlling player) has to put the card into its owner's archive.
Whether Biomatrix Backup text is actually in "direct contradiction" with DESTROYED is (imo) up to one's interpretation to what constitutes "in direct contradiction". If you go with the (admittedly low standard) of "As long as you can follow both the rules and what is written on the card, you have to do so" I'd argue there is none. You can both observe DESTROYED's and Biomatrix Backup's text, and would then be forced to archive the card. (With this low Standard there is no contradiction, as both Biomatrix Backup text and DESTROYED actually allows you to so, DESTROYED is only in conflict with the possibility of [whomever] choosing not to archive).
In this interpretation Biomatrix Backup is not actually in conflict at all with ACTIVE PLAYER.
b) Invoke the GOLDEN RULE to ignore ACTIVE PLAYER and DESTROYED rules: (If you think Biometric Matrix' text is in "direct contradiction" with both ACTIVE PLAYER and DESTROYED):
Whether Biomatrix Backup is in "direct contradiction" with ACTIVE PLAYER is again (imo) up to the interpration of "in direct contradiction". If your Standard of "in direct contradiction" means that a card allowing "you" (and as i have stated before, imo every interpretation of "you" not being the controller of the card leads to ridiculous results) to choose something ("you may") and ACTIVE player disallowing exactly that thing are "in direct contradiction", would mean the GOLDEN RULE would give the text on Biomatrix Backup priority over the rules governing ACTIVE PLAYER. So despite ACTIVE PLAYER forbidding it, the controller may still archive because the GOLDEN RULE reigns surpreme.
In this case Biomatrix Backup is also arguably in "direct contradiction" to DESTROYED as you may choose something in contrast to abilities resolving automatically. So the GOLDEN RULE would have to be invoked in this instance to give priority to the cards text as well.
c) Invoke the GOLDEN RULE to ignore DESTROYED rules, but resolve in accordance with ACTIVE PLAYER rules:
I thankfully don't have to really write up what this does, as this is the interpretation that (imo) FFG has taken (even if it was not openly mentioned) and you can read their reasoning in the FAQ.
In summary and as a TL,DR, i think that FFG has taken a sensible (and possible) interpretation of the rules as written, even if it clashes somewhat with the framework (of DESTROYED) and is rather unintuitive. While there is the possibility (imo at least)to interpret the rules in a way that conforms with all Glossary definitions of terms (variant a , I'm not particular convinced this isn't stretching not beeing in "direct conflict" even if this seems to have been the original intent by Richard Garfield).
I also wanted to point out, that while a lot of people have rationalised and probably even supported FFG's ruling as to not have to invoke the GOLDEN RULE, I actually think that the GOLDEN RULE is already used to ignore DESTROYED (even if this is not adressed directly).
So while i think FFGs interpretation is a possible one, if somewhat unintuitive, i disagree with Richard Garfield and FFG (and probably a lot of other people) that the way this was ruled was by necessity the only way the rules could be read/interpreted (something that at least to me Richard Garfield's post seemed to imply.)
As a sidenote, i think it is actually necessary to define the "you" player in the Glossary. While i firmly believe context and otherwise absurd interpretations of cards already make it clear who "you" refers to, the absolute absence of a term quite important to and used on so many cards is simply baffling. I also wouldn't mind a longer explanation what actually constitutes beeing "in direct contradiction" with the rules.