Older CRTs were convex because that's the only way CRTs can emit photons (from a cylinder). A concave display could reduce glare because less of the surface is facing a light source. A convex display has more glare because more of the surface faces a light source.
We all know why old CRT's were curved. Your explanation is weird, and wrong. But you apparently don't know about the first flat display, which was a Zenith CRT monitor. The benefits were immediately obvious.
The reason why old CRT's were convex, and by the way they weren't like a portion of a tube, that came later with some Sony displays, was because the electron beam spreads as it moves to the edge of the display. It turns into an eclipse. Until electronics were able to keep the beam shape round, they had to have the front of the screen shaped like a portion of a sphere, but flatter. Zenith was able to correct the beam distortion, and so could use a flat tube. Sony's screens, which came earlier than the zenith, used screens that were straight vertically, but still curved horizontally, because it was easier, as the beam had further to travel to the edge lengthwise than height wise.
Now, as for your concave idea, well, it's terrible. A concave surface will collect glare from every angle. It will be far worse.
And so there you have it.
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That was because the older CRT monitors all had a CONVEX surface, which indeed increased glare. Relative to a convex surface, a flat screen reduces glare. But the OP said if the surface could be made just slightly concave, it surely would further reduce the glare, which is very true physic that you totally missed.
No. Making the screen concave will increase glare significantly. If you ever get hold of a magnifying mirror, used for makeup, you will see that it collects light. It's shaped like a telescope mirror. You really don't want a screen shaped like that.
I only minored in physics and had just one year of optics, so I suppose I could be wrong, but I highly doubt it.