With respect to the shirtsleeve environment of TMP Main Engineering, the intermix chamber couldn't have been radiating any appreciable amount of power by its glow, relatively speaking, or the engineers would have been killed on the spot. In fact, that it was only as bright as it was, and not vaporizing the people in there with it, means that the shielding on the intermix chamber had to be pretty darn effective, >99.99%.
Well, yes, that's the obvious handwave, and in my post-TMP novel Ex Machina
I asserted that the actual intermix happened in a very narrow tube in the middle of the shaft and the rest was just a very, very thick wall that blocked most of it.
But realistically, any reactor is going to produce waste heat, and for a reactor that powerful, it would be far more sensible to put it at some distance from the inhabited portions of the ship, with no intervening atmosphere to transmit that heat. You'd want the reactor on the outside of the ship and connected directly to heat radiators to bleed the waste heat into space. Indeed, that was Matt Jefferies's original intention in putting the nacelles out on struts away from the ship -- to protect the crew from the engines' heat and radiation. TMP's decision to put the warp reactor right in the heart of the ship was a retcon I've never been happy with. (Although admittedly TOS set the precedent when it put the dilithium chamber right in the middle of the engine room set.)
Why it was even necessary to see the reaction occurring is another issue; perhaps being able to see the reaction was important to gauging and regulating engine performance, in-universe. Even Forbidden Planet allowed the Krell to view their nuclear reactions, via a special mirror.
There's no reason it couldn't have been monitored remotely. As stated, it's just a fictional contrivance, arising from the fact that these things are designed based on what looks cool rather than what makes functional sense.