Just throwing another diagnosis here for perhaps installment #4.
I believe healing is one of the culprits to slow combat. Two leader words and an emergency button is all you should need for a party of 5. For a party of 6, add one more emergency button.
Healing is not fun to me. Learning to distribute damage taken among the party reduces the amount of mid-encounter healing, and during a short rest you can do all the post-encounter healing you need. I find this to be a fun tactical challenge.
Healing vs damage taken is often an arms race between DM and players. The more healing you have the more the DM wants to hurt you to challenge you, and either encounters run longer, or because of massive damage monsters have to do, you either run low on surges fast, or end up with some close calls or a few deaths due to the impact of focused fire from massive damage dealing monsters, especially against parties of 6.
The less healing you have, the DM will adjust the damage coming at you accordingly, and you can have thrilling combats, avoid slug fests, and can play through more encounters a day.
With the current state of things, it's not unusual for me to see a defender burn through 6 out of 12 surges in 1 encounter, and a striker burn through 4 out of 8. This is because there are so many ways to trigger healing surges mid-encounter, and you still have to heal up at the end. I would much rather see the defender use 3-4 surges per encounter, and expect the rest to use 2-3 surges (depending on their build and role).
I've played and run encounters where there was no leader in the party, and they are absolutely thrilling, and they are fast. I feel healing slows down combat (via DM meta to challenge PC's) and should be considered a culprit when examining solutions for speeding up combat encounters.
Why are low level encounters fast and furious? Because there aren't as many HP resources and healing surge triggers, and because monsters deal a proportionately challenging amount of damage. At levels 1-4 I was able to run 3 encounters in a session easy with a good bit of roleplay in between, and 5-6 encounters in an adventuring day. As you get to paragon, I don't plan for more than 1 combat encounter a session (so we actually have some time for RP), and I have a difficult time running more than 3 encounters an adventuring day.
When examining all the knobs and dials to speed up encounters, I think it would be prudent to take a look at the healing faucet as well.