The idea behind it was likely inspired by WOW's Holy Priest and their Holy Words CD reduction system.
WOW examples:
For example, Casting Flash Heal or Heal (Cure1/2) reduces the CD on Power Word Serenity (Tetra). Casting Smite (Glare) reduces the CD on Power Word Chastise (1min CD damage with a stun, we don't have an equivalent). Casting Prayer of Healing (kinda like a targetted Medica1, but a lot weaker) or Renew (Regen) reduces the CD on Power Word Sanctify (ground targetted Cure3, bit of an odd one).
But then the real depth in the WOW version of the system is how there's synergies on top of those interactions. Cosmic Ripple is a talent that causes an Assize-like burst of healing emit from you whenever the CD on PW Serenity or PW Sanctify finishes, incentivizing you to keep using them, and then using the things that make them come back faster. Empyrean Blaze causes your Chastise to empower your Holy Fire (a second, 7s long super-strong DOT) to skip its 10s CD and allows it to stack multiple times on the same enemy, allowing you to deal good burst damage with it (if it's safe to spend GCDs on applying it). Revitalizing Prayers causes Prayer of Healing (the scuffed Cure3) to have a chance to also apply Regen to the allies it hits, and those extra Regen procs also count as 'casts' so they also trigger the CDR on Sanctify.
And the biggest one is Power Word Salvation, a TWELVE minute CD that, in FF terms, heals like a Medica1, applies 2 stacks of Excogitation and one stack of Regen, and does this to the entire raid (eg in an Alliance Raid it'd hit all 24 players). Each time you use Power Word Serenity or Sanctify, the CD of Salvation is reduced by 15s. So it's a CDR system on top of a CDR system, and this means that the CD is effectively closer to 4min rather than the full 12min. But because the action has such a long native CD, it's allowed to have such a crazy sounding effect, which increases how impactful it feels when you do use it.
I think that the SB system wasn't a bad idea, in itself. It had implementation issues. First was SE's fear of attaching anything interesting to the damage buttons. We could have had, for example, 'Fluid Aura (if they hadn't removed the damage from it) reduces the CD on Tetra by 2s', or 'Aero 2/Aero 3 DOT ticks decrease the CD of Benediction', but instead everything was hamfisted onto the healing GCDs. Second, was that they tied every Lily spending action, to the same resource (the Lilies), causing the player to feel like they have less agency in what they want the CDR to apply to. It became more a case of 'I use this because I need it now, and if it happens to get a CDR, so be it'. You never used Tetra 'because I want to make use of the CDR of the Lilies', you used it because you needed it at that time, and the Lilies happen to trigger on it. And because the game's combat is so scripted, the CDR on these healing tools was ultimately 'un-noticeable', if you had Asylum planned out to be used at 0:30s, and at 2:30s, it didn't matter if its CD was reduced or not, you were going to leave it ready to use until 2:30 either way. You could literally hide the Job Gauge from your UI and still play just fine, because the system was so... ignoreable. And thirdly, because everything was tied to the same Lily gauge, one 'choice' (if you can call it a choice, since everything auto-triggered the effect if it was available to use) became THE choice: Assize. Because it did damage and the others did not.
Which just goes back to the old talking point: SE's insistence on trying to show that Healers don't need to press damage buttons for any reason ever, is closing a lot of design options off. Rather than trying to make designs that allow for Healers to simply ignore their damage buttons, wouldn't it make far more sense to have those damage buttons (and their interactions) ultimately be 'unrequired' to clear most content (via intelligently balancing their potencies etc.), and then add those small interactions that incentivize (without forcing) the player to at least experiment with dealing damage? For example, if there's a simple interaction of 'Stone/Glare extends the duration of Asylum by 1s', wouldn't that incentivise a player who's normally deathly scared of letting people die to think 'oh, Asylum is down so it's probably safe for a little while, and if I use Stone I can make the Asylum (and therefore the safety time) last longer' at least a little bit?
The objective with Healer design should be to get players to try dealing damage, at least once. Because overcoming the mental barrier and going from zero damage spells, to at least one, is a bigger challenge to mentally overcome compared to any amount beyond that. If you can get a player to start using damage abilities even a little, it would reveal to them how little healing is actually required and how much time there actually is to deal damage, when people aren't standing in orange. And then that opens the way to them saying 'okay, there's actually a lot more time for me to use these damage buttons, and take advantage of their interactions (like extending Asylum, reducing Tetra CD, etc) than I first thought'.