Only in very specific 'cathode ray tubes' (which is how traditionally monitors and TVs would work) do the particles become a beam.
Can you unpack "speed" there?
I figured the speed of an electron in an orbital was the same as the speed of the atom.
Certainly there's some energy associated with it being in that particular orbital, and if it takes on more energy then the region in which it might be measured will expand, but it's not like zooming around in there is it?
https://en.wikipedia.org/wiki/12AX7
https://www.effectrode.com/knowledge-base/the-12ax7-tube/
The suggestion of the parent comment to use a nano-sized particle accelerator on a chip could somehow be useful for tube amplifiers just seemed ridiculous.
Just to nit pick your "same speed" comment though...
The tube you linked to indicates 250V at the anode. Electrons have a mass of 10^-13 kg. For such a tiny thing, that voltage is positively tremendous. Even in the tiny space of the tube I think that the electrons are accelerating from 0 when at the cathode to relativistic speeds at the anode. The tube is an accelerator, and the control grid opposes that acceleration.
Accelerating the electrons to even greater energies would just mean that more voltage is needed at the triodes to have any modulating effect on the signal. Also, the beam would be, well, a beam, rather than a... gust? It would either strike the anode or miss it--that's not modulating anything, that's just an on-off switch.