The problem isn't just that tanks self-heal too much right now (especially WAR), it's also that healers and tanks are fundamentally no longer designed to actually work together at all.
As the poster above mentioned, healers do so much of their healing with instant abilities or passively that it's very difficult to impossible for a tank to be aware of what a healer has up at a given time beyond Benediction (two sentinels per Bene, at least that's easy). The only thing a tank can really know is if their healer feels comfortable enough to be casting DPS spells, and even those have been pared down so hard you have SCH sitting there with 3 of their 4 DPS abilities instant cast, or if they're chain-casting their most expensive spell in a panic. Usually I'm just like "Ok, I push DPS button 1-2 1-2 andturn to the left sometimesput up a defensive cooldown sometimes." Clemency is effectively useless, 99% of the time you're going to get a Tetragammaton or something when you're wasting time casting it and you've pissed off the WHM. How exactly are you supposed to know that's coming? Even Equilibrium, I find I overheal a lot of the time because there's just enough lag between when I use it and my HP pops up that the healer has used one of their abilities on me too.
The same is true to a slightly lesser degree with the tank skills. Like WAR's utility used to be Mantra, which worked in synergy with healers and stacked with Convalescence, on PLD I used to Stoneskin the party top down while WHM did bottom up before a raid ultimate AoE or time a cast to land when I would get a Disease debuff. Now it's just "push button, shield party."
Plus hate has been buffed so much, it's basically impossible for a healer to rip from a tank now. It used to be a PLD had to juggle keeping Blind up from Flash as long as possible and managing hate from the WHM or AoE damage, now you just 1-2, 1-2 and Requiescat sometimes (actually, I start with my caster phase in dungeons so it's more like 1-1-1-1-2-2-2-2)
The only things really left for tanks and healers to coordinate are alternating protective buffs on AoE damage, which is unnecessary for 90+% of content, and holding off big defensive cooldowns while stun is up from Holy if you're running a dungeon with WHM.