Depends on the type of tape used. Normal insulating tape can become brittle and fall away in time. I used to use choc-blocks almost exclusively, but I'm a Wago convert now, especially the re-usable type with levers to clamp the wires down.

I wish you hadn't told me that 

I guess it's all a matter of preference and what the local codes are, as well as what the local electricians use.
Wirenuts are just like anything else, safe as long as properly installed.
I've seen my brother, Morbid Angel, who is an electrician take 4 #12 wires twist them tightly together with his Kleins then twist on a wirenut.
He manages to make the connection so tight it's as if the wires were never cut to begin with.
For safety before he shoves them back into the box he gives the wires and the wirenut a gentle tug to make sure all of the wires are tight and checks to makes sure that no bare copper is exposed below the skirt of the wirenut.
Installed in this manner he says tape is absolutely unnecessary.
In fact he cusses very loud when he finds taped wirenuts.
Aside from the sticky mess he knows that taped wirenuts usually aren't properly installed.

Personally I prefer using the wirenuts here but the Wagos with the levers do sound nice for more temporary stuff I don't intend to keep wired up for long. And yeah if you properly install a wirenut they're pretty secure and you shouldn't have to use tape. Like WigWag said, give them a gentle tug to make sure the wires are snug and check to ensure nothing is exposed. In fact I don't like tape since it gums up everything after it dries.

I sometimes reuse the fluorescent sockets too. You're technically not supposed to but I find if you gently pull while turning the wire you can usually release the wire without bending the spring wire clip. The wire will be a bit nicked when you pull it out so you might want to clip it off and restrip. If you find them not holding well to a new wire it means the wire clip got bent and you probably should replace the socket. 

When I have looked at some 1910s wiring, I have noticed that exposed connections consisted of exposed wires twisted together and wrapped up in tape rather than using wirenuts. On one of my Jefferson 400w H33 mercury vapor CWA ballasts, the instructions say to “tape off unused leads” rather to cap them off.

I guess it's all a matter of preference and what the local codes are, as well as what the local electricians use.
Wirenuts are just like anything else, safe as long as properly installed.
I've seen my brother, Morbid Angel, who is an electrician take 4 #12 wires twist them tightly together with his Kleins then twist on a wirenut.
He manages to make the connection so tight it's as if the wires were never cut to begin with.
For safety before he shoves them back into the box he gives the wires and the wirenut a gentle tug to make sure all of the wires are tight and checks to makes sure that no bare copper is exposed below the skirt of the wirenut.
Installed in this manner he says tape is absolutely unnecessary.
In fact he cusses very loud when he finds taped wirenuts.
Aside from the sticky mess he knows that taped wirenuts usually aren't properly installed.

I would bet he used something like a Marrette 35 for these four wires!

Otherwise you've pretty much described how I make connections with wirenuts! Back when I was in that industry myself I also made sure to position the wirenuts skirt down inside the box so that if for any reason they got loose, they wouldn't fall off the twisted wires.

And I also strongly dislike electrical tape on wirenuts LOL.

When I have looked at some 1910s wiring, I have noticed that exposed connections consisted of exposed wires twisted together and wrapped up in tape rather than using wirenuts. On one of my Jefferson 400w H33 mercury vapor CWA ballasts, the instructions say to “tape off unused leads” rather to cap them off.

That was back in the knob & tube era. The industry standard was very different. Connections were usually tin-soldered then taped. If undisturbed, K&T wiring is very reliable because of this. The connections just can't get loose.

The 1910s wiring installed correctly probably used one of the various splicing techniques at the time. Wires were twisted together in such a way where pulling on the splice will cause the splice to tighten instead of loosen. Solder was then applied to prevent the splice from unraveling. Finally the joint was insulated using a rubber-cloth type of electrical tape also known as friction tape. These types of splices when done correctly were pretty robust but rather labour and time intensive compared to later methods such as set screw marrettes, screw terminals, wire nuts, and now Wagos.

IMO using electrical tape on most splices is a older practice anyway and with modern wirenuts it's more often better to avoid using tape if you can. Your ballast is probably an older one since it mentioned taping off unused leads. The modern practice would be to cap off with a wire nut, cap with a crimp on connector, or cap off with heat shrink.     

If you really must use tape, self-amalgamating tape won't get sticky, although it will degrade if exposed to sunlight, due to the UV.

I usually use heat shrink tube instead of tape for most applications. It looks neater and is a lot easier to remove as you can just cut it off.

That reminds me, I've got to get some larger diameter heat shrink tubing, the stuff I've got isn't big enough to go over some things.