Yes, watch both. The Cage flows better watching it straight through rather than being inturrupted by the part split and regular cast commentary. Also, the ending is great. It's a nice look into the Trek that Could Have Been.
However, to discount The Menagerie is a mistake. I rarely sit through The Cage portions, or even the second itself, but the first 40 minutes of The Menagerie are really good. The tone is so grim and serious with great dialog and performances. It feels like the last of the pure Roddenberry scripts, as his stuff was a lot more serious overall than Coon's. The stuff between Kirk and McCoy is outstandingly written and acted, with the awesome turnabout of McCoy taking a stand defending Spock againt Kirk. It's also beautifully ironic because Bones is wrong.
The ending raises more questions than it answers (the Mendez illusion and so on), but the last minute of Kirk watching Pike restored to happiness via illusion is very touching. The endings of both can be reconciled without much effort (Pike saw Vina leave with his illusional doppelganger; 13 years later, he returned to live the scene himself) and they both add to the rich historical Trek tapestry.
Until a time machine is invented that can send Okuda back to the late 1960s and he is lucky enough to get a job at TOS while there, he is no more than you or I... a TOS fan. Okuda is no more official than you are in my book. He's a nice guy who didn't work on TOS.
As for Jefferies... he knew the issues, and didn't have final say. But he never (ever) considered the exterior turbolift tube to be anything else. And when given the chance to revisit the design, he added a second one (rather than try to do a LiS solution or moving the bridge somewhere else).
Did he like the angled bridge? No. But he lived with it.
Do you have something to show that Jeffries felt this way? Because then by your own admission you have no more credibility than anyone else not working on the series. You can believe what you want, but don't try to pass it off as fact.
Honestly, I have no idea if the bridge is angled and that structure is the turbolift, but if it is, then there was no "'tween hull" designed beyond the controls and the turboshaft. No room for machinery or hull protection. So then the viewscreen butts up against the front wall, just to the right (or left depending on what direction you're looking) or that little white rectangle on the outside. Not that the rectangle is supposed to be anything other than a white rectangle, but why create the illusion of a window/screen up front if it's not going to be there? It's more likely that there wasn't that much thought given to where everything sits in relation to the miniature, and if they did spot a discrepancy, then they probably figured nobody would notice. It's not like they focused on that part of the hull. This happens on TV a lot, especially when dealing with special effects and miniatures. See the Jupiter 2 and Seaview sets for some vivid examples of this, and the designers were no less talented than Jeffries.
I always thought it was pretty stupid to put an elevator shaft sticking out of the hull like that. It's not even a scenic elevator, so what's the point?
So unless you can provide something from Jeffries or Roddenberry stating that the structure is the turbolift and that the bridge is weirdly angled, and that Jeffries just tolerated it, your fan speculation is just as valid as any other.