None of the original versions of the support items had passive gold generation. It was added after they discovered that RNG CAN happen and this is a very simple solution in order to smooth out the gold generation curve. Basically you can reduce the amount of gold generated from the actual passives because you are getting passive gold. Its not a hard sell and its also really really easy to understand.
Your proposal promotes the auto attacking of minions and then having your ADC last hit them. It does not matter what you call it, honestly, you can call it whatever you want. That in and of itself is the reason your not selling your idea and why I said it is bad. Why should MY gold generation be based on how well my duo can CS?
( I had information here that i thought was right but was incorrect)
Now if you want to be really technical about it the original coin granted HP and gold on minion death and had nothing to do with anything but them dying. It was OP on champion like Soraka and it let you do really stupid things in lane and not be punished for it. Not to mention how a support's auto attacking of the wave can and almost certainly will throw my adc's timing off while he is trying to secure CS if I don't IMMEDIATELY attack all the minions and this little bit of damage will STILL wind up shoving the wave if the enemy support runs a different gold generator.
For example Janna at lvl 1 has something like 45 AD, if she decides to auto each minion, while the enemy support (say a poke based champ with Spell thief's) does not then she will do a little less than 300 damage to that wave at level 1. Roughly 300 damage across all minions WILL push a lane eventually (ever wonder how those super waves of minions are created). That is my point and that is why your idea is not viable.
It honestly feels like your just upset that a heal bot can actually babysit their teammate. If your having difficulties vs Soraka just poke her first and whenever she walks into the minions to guess what? collect COINS. Its not brain surgery. For your other champion who 'does nothing but heal' Sona actually would prefer to use the spell thief's edge because last I checked she loved to run in (sometimes with e) then q auto with her passive then pop her heal to heal up some of the damage she's OBVIOUSLY going to eat. She has only calmed down because with the way ADC and support mastery/itemization is. It makes her poke pattern ineffective (adcs have much more sustain thanks to warlords and supports generally being more defensively oriented in response to how powerful ardent censor is). IF ardent censor wasn't such a HUGE power swing currently for as cheap as it is and adcs didn't regenerate as quickly Sona will probably still use that aggressive play pattern where applicable.
FWIW : The current RNG on the coin is probably less RNG than your trying to sell it as honestly, if I were to hazard a guess on how it works under the hood so to speak, I would guess its similar to an adjusted weight type RNG (where basically you set the drop % of each coin then as one drops more than than the other based on the set % the chances of you getting the other one increases). This in combination of the passive gold generation does exactly what I said previously, smooths out the gold generation to minimize gold gen fluctuation as well as allowing the supports to actually go make plays.
Old coin encouraged the support to never leave the lane... ever because they lose a lot of gold that way. New coin eases that immediate demand to sit in lane all day under tower by having there be coin pickups in lane after a few minion deaths that are not obnoxious while also not harming the supports desire to do things like go ward somewhere.
+5% cooldown reduction
+50% base health regeneration
+10% cooldown reduction
