The cogwheel cursor appears, but clicking on the orb does nothing. No channel bar appears and I’m not given control over the boss. Adds spawn as normal and the boss wipes the room upon death with eggs up as expected.
It’s been a long time since I tried to solo BWL, but this has definitely worked before.
(Why even force max-level characters to do fiddly mechanics like this while soloing a 20-year-old raid, anyway? Just let me kill him and move on.)
Were you doing this on your hunter?
Did you still have your pet out?
Your description matches what happens when a pet class doesn’t dismiss their pet before trying to control Razorgore, because the dungeon is so ancient they didn’t have any better mechanic for the encounter than borrowing the Eyes of the Beast spell. Which won’t work if you have your own pet out.
Editing to add: yes, I learned this lesson the hard way. My suggestion comes from personal acquaintance with the situation I described. 
That raid had been tricky ever since we were able to solo them. 20 years ago the devs never imagined a need for a working solo mechanics. When they tried to tweak the mechanics a few expansions later, they didn’t have the original team anymore and had to muck around the code a bit to see what can work without breaking the original raid mechanics.
If dismissing the pet doesn’t work, try turning off all addons. I just did this as a druid with no pet but some 50 addons and after some slow start, I was able to click on the orb and destroyed all the eggs.
Oooh, good call. Didn’t even think of that. I’ll give it a go next time I’m in there and report back here.
The devs did do one soloability nerf way back in Patch 7.2.5: making “Destroy Egg” instant-cast instead of a 3-second cast, so only one MC phase is necessary if you’re reasonably quick and efficient.
I was farming this before they did that. It was like having the world lifted off my shoulders.
I’ll save you the time by confirming what Sniperorc said. Just took my hunter alt out to BWL and sure enough I couldn’t control Razorgore unless I dismissed my pet first.
Legacy spaghetti. ¯\_(ツ)_/¯
Yep, just confirmed it myself as well.
I should make spaghetti for dinner to celebrate.