CRTs do have pixels. If they didn't, you could run an SVGA signal (800x600 at 60 Hz) directly into any CRT. If you tried this, it would likely damage the tube beyond repair.
I certainly saw aliasing problems on CRTs, though usually on computer monitors that had higher resolution and better connection standards. The image being inherently "soft" is related to limited resolution and shitty connections. SCART with RGB connections will bring out all the jagginess. The exact same display running on composite will soften it and make it go away, but at the cost of a lot of other things looking like shit.
CRTs do have pixels. If they didn’t, you could run an SVGA signal (800x600 at 60 Hz) directly into any CRT. If you tried this, it would likely damage the tube beyond repair.
Would it, though? I'm skeptical.
If it did, it wouldn't be because they have "pixels," though; it would be because overdriving the deflection yoke with higher-frequency signals would generate too much heat for the TV to handle.
Otherwise (if it didn't overheat), it should "work." The result might look weird if the modulation of the signal didn't line up with the apertures in the shadow mask right, but I don't see any reason why sweeping the beam across faster would damage the phosphors. (Also, I'm not convinced a black & white TV would have any problem at all.)
It will tend to turn the beam on when it's off to the side, outside the normal range of the screen. X Windows users in the mid 90s had to put in their exact scanline information or else the screen could blow up. That went away with a combination of multiscan monitors and monitors being able to communicate their preferred settings, but those came pretty late in the CRT era.
Edit: in any case, color screens need to have at least bands of red/green/blue phosphor. At a minimum, there will be breaks along either the horizontal or vertical lines, if not both.
You had to tell it when to change lines, when to start firing on a new line (i.e. you changed lines then waited a bit and only after started sending data), then when to stop firing and finally another wait after which a change lines (so there was some "empty" at the start and at the end of a horizontal electron gun trace and you had to tune those so that the image didn't start or end outside the screen).
There was also something similar for the vertical axis - i.e. instructing it to go back to the top and some empty lines at the top and the bottom.
I wouldn't say it was convenient (it was a bloody text file with weird-looking numbers and you did run the risk of blowing up the CRT), but it was kinda fun that you could create your own crazy screen resolutions for X once you understood the principle of the thing.
When you say "blow up" do you mean the tube would literally explode, it would burn through phosphors, a circuit board would let the magic smoke out, or something else?
I remember configuring mode lines in X. Luckily, I never found out the hard way what happened if you got it wrong.