So I charged my 3D up to 100%. The MHL converter has held the battery at 100% through two HBO episodes of the Sopranos. I'm seeing your results now EarlyMon. I'm thinking the same thing should work for the One.
Here's a helpful data point.
One of my video subscriptions works better with a good browser in desktop mode than with their app.
One of my browsers feeds that with one core running at 1.13 GHz, with the other core sometimes hitting 384 MHz. Another browser runs the same site, same job, same results with 2 cores drawing 1.512 GHz.
What can I conclude? The second browser was written poorly. It sucks resources no matter what, accomplishing nothing. It's not smoother, faster or better, it's just a little bit of a piggy when it gets the chance on some sites.
So, let's look at your 3D results.
Unless your HBO experience is lacking and I missed that, then what I heard was a Snapdragon S3 dual core on your 3D was plenty capable for HBO.
Going to the One, I'd expect less of a system load overall - your phone screen is already 1080p, so there ought to be no additional software paths, say through the framework, managing different resolutions, one for the TV and one for the qHD display as happens with the 3D, and if you turn brightness down, the additional display control circuitry for full HD on your One display ought not come close to eating up the processor savings.
And yet, the HBO app is consuming more resources to do the same or less required processing from a systems perspective.
So I'm inclined to verify what you've already thought from your Netflix comparison - the HBO app is being a piggy.
I don't think that a theoretically better MHL rig could really solve this. Once programmers get it in their heads to write bad code, they can take any hardware to its knees.
I don't know if this exists for your model, but if I were to try to solve this with root, I'd look to see if there isn't a governor you might switch on to force your phone into dual core mode only. (The hardware would be Ok with that, the CPUs are made to scale up and down and kick in when needed.) Or a clock limiting setup. (I'm not a big fan of setcpu, but you know it well and could work here. Rooted, pretty sure you could do this with Android Tuner for example.)
In other words, something you switch in when you're using HBO that will limit resources and therefore limit power drain.
Your 3D proves that it doesn't take as much as a One running at full blast to get the job done.
That's my opinion.
PS, I'd like to address this as well -
I hope it's just a software thing. I wonder if people have tvs that does not put out enough power via MHL to the phone. If that's the case, I would suppose an update could resolve the problem on a tv.
If you mean for HDMI, please see -
http://www.hdmi.org/learningcenter/kb.aspx?c=13#42
If you mean an MHL-equipped TV, then my guess would be that like anything else in the TV business, they only provide for the spec at the time. Each revision of MHL has been accompanied by new MHL controller chips, fwiw.