Ah, the projectile hijinks of youth, what memories.

I too started with Lego, though my effort was of a much simpler design than the 1 in the video. I used a combination of ordinary & technical Lego, & heavy duty elastic bands, & in operation it bore more similarity to a crossbow than a pistol.
Other things I tried included a coathanger catapult, & something which I never came up with a name for, but fired the ink tube & nib of a biro from a pen barrel.
The coathanger catapult was simple. I used 1 of those wooden coathangers that had these little hooks underneath [dunno what they were for]. I looped 3 elastic bands together & strung them between the 2 hooks, & used solid core wire as the ammo. I took a piece of solid core wire a few inches long, then folded it in half repeatedly until I got a C shape about an inch in diameter. Hook that over the elastic bands, pull back & release; good for at least a 100 yards.
The pen 1 requires 2 different types of ball point pen: the ones with the conical shaped nib [Bic, I think] for the barrel, & another type for the projectile, though I don't know name or brand. You used to be able to get them with a transparent barrel tinted to match the colour of the ink. The ink tube with attached nib could be removed by unscrewing the other end. The nib was a thin, straight shape.
To make the pen projectile device:
- Disassemble the Bic, so that you have just the barrel.
- Do the same with the other pen, but this time you want the ink tube [with nib still attached].
- With 1 hand, firmly hold 1 end of an elastic band [band width should be at least 5mm] against the barrel of the Bic, about 1/4 of the length distance from the wider end, & stretch the other end of the band over the open nib end of the barrel.
- Drop the back end [not nib] of the ink tube from the other pen into the open end of the Bic barrel, so that the nib of the other pen is pointing up.
- The ink tube should drop down until it stops against the band covering the nib opening of the Bic barrel.
- With your free hand, stretch the band further away from your other hand. This should allow the ink tube to come through the Bic nib hole, & you should be able to grasp it with the band.
- Aim, pull back & release.

Crude, but simple. In my middle school, the assembly hall used to have some tables at the back, where I used to sit & read, to keep out of the way of the bullies during lunch break [they'd usually be outside playing football or doing something else just as boring

], & because I wasn't really interested in socialising with the other kids. I reckon the ceiling of the hall must have been at least 40 or 50 feet high, & by the time I left that school there were several pen ink tubes embedded in the ceiling tiles. I went back out of school hours a couple of years later, & peered in through the window - they were still there!
Another, simpler, projectile device I just remembered consisted of a straight rod or bamboo cane & a small apple [or small potato]. All you do is push the apple or whatever down onto the rod/cane, & slide it down a bit [it's been years, so if you want to try it you'll have to experiment], then swing the rod/cane over-head in a similar motion to that of the arm of a seige catapult. Distance projectile will go depends on length of rod/cane, how far down you pushed the projectile, how hard you swing the rod/cane, & the weight & size of the projectile.
I did briefly experiment with a proper air pistol, but I won't go into that as it's a bit of a sore subject with Dilli.