There are a few problems with these sentiments:
1) The only effective time to use Diversion is in the opener.
Mid-fight Lucid is way better at handling your aggro, massively so, because it cuts all the aggro you've generated so far in half, including on subsequent uses. Popping Lucid about a minute into the fight (typically 45s due to the next reason below) is about equal in value to Diversion in the opener. And only gains in value for each use afterward.
2) Diversion disrupts DPS openers if you actually wanted to use it for full value.
Diversion is 90% reduction for 12-16 GCDs assuming you were popping it right before starting the fight or in the first GCD you use, in practice it's way less than that on some classes because of what they need to do pre-pull. RDM specifically misses about 5-6 seconds of it. In most cases you can't use it in an ideal manner to achieve that in the opener that doesn't disrupt your oGCD rotation in some way, making it absolute garbage.
3) Diversion isn't going to magically fix bad players or systemic issues.
Any DPS complaining that the tank isn't doing their job on aggro past their levelling experience is usually not doing their job to begin with and is likely in a roulette which doesn't really matter. I've used both of those buttons in dungeons every pull and still can rip aggro despite that. Why? Because of gear disparity, and I've seen healers rip aggro off of tanks just as easily for similar reasons.
4) Giving tanks more aggro out of a tool they already want to use is more intuitive and less disruptive. Aggro is a tank-specific mechanic. Tanks should have to care about aggro.
Shield Lob and the like are already used for pulling mobs. This change actually ensures its available every single pull. Having to pop it again mid-fight isn't an issue for most people learning in tank stance while undergeared. Circle Shirks and tank swaps are already covered. Only PLD and DRK lose out, as both would need their tank stance up to take advantage of this and thus need to clip their GCD, which works fine for dungeons but sucks in aggro-reset situations for them specifically. All tank stances should be oGCD based anyways. From a Savage perspective the tools non-tanks have for the most part can enable tanks to pull in DPS stance (though it's most often the WAR in tank stance because of Unchained specifically getting around the problem) which is not okay. Tanks need to work for their aggro more in general, but giving them a tool to circumvent their problems in the opener specifically is far more efficient in terms of design space and in practical terms.
5) Any DPS that still has issues has been addressed by the Invigorate change I mentioned.
Melee DPS do not use Purification, Elusive Jump, or Merciful Eyes for aggro reduction unless they absolutely have to. NIN has Shadewalker to literally donate their aggro to a tank. They all lack a mid-fight way to deal with aggro that doesn't disrupt their rotation in some way (MNK/SAM lose damage, DRG is literally put out of melee range) apart from NIN, meaning they really should get the aggro cut Lucid has to deal with it. Hence the Invigorate change. It solves the problem in a much better way than Diversion or their personal options ever will, and using those options is a necessary evil in the interim. Don't even get me started on BRD/MCH. Their problem is the lack of Diversion in the opener, which is also addressed with the tank change I offered.
In your case with RDM, I can only assume you're not using Lucid on cooldown or early enough if you need to use Diversion more than once a fight. Find someone to Mana Shift early to get it rolling faster if you don't want to 'waste' MP, and never let it sit on cooldown after the fact. You'll be fine if you can get through the first melee combo. And if you can't for some reason, check your tank's gear.
There are classes that need better aggro tools, but they're not DPS, they're healers, and of them only one is an actual issue right now. WHM has no real aggro-reduction or cut tool aside from Lucid but has some of the highest aggro generating spells. Keep in mind AoE heals are like a 4-8x aggro boost since every enemy cares about who is getting healed but damage wise enemies only care about the damage dealt to themselves for aggro purposes. AST has free permanent aggro reduction in their Sects and Star doesn't generate any at all. SCH gets to split their aggro with their fairy. To me Thin Air & PoM would be good candidates for a Diversion-like effect that would benefit them in particular.