Ahhh.. the irony of Electrical Tape. Its neither electrical (not having any specific insulative value), or sticky (like tape).

Just go to Ideal electrical of Lawrence and Hansen or similar and get a good quality electrical tape, the one that sparkies use. Nitto is good but has its own name embossed down the center which may not be what you're after. It won't be sticky like you hoped, but there is more of a skill to wrapping then anything else including sticky. Start off, but wrap over yourself once or twice to wrap the end down before you then move along the loom. When you get to the other end, wrap carefully, and tuck the end into the wiring harness, (through the wires). This will stop it from unraveling. Also, wrap the legs off first, before you do the main harness.

I prefer woven loom wrap or other methods over tape any day, but if you're after the original look you will be fine. My mk2 loom is 11 years since re-wrapped in good tape and it still looks great.

I feel you are describing the wrapping of non-adhesive type tape, which I used on the Mk1 recently. I also used an excellent cloth tape under the non-adhesive cover.
But I want to copy the later style, where the tape is only used in sections and not continuous.
I have used Nitto many years ago - made in Japan - and it was excellent, but now there are many versions that are not so excellent.
If the worst comes to it, I will use the cloth tape I bought from UK (Vehicle Wiring Products).
Thanks for your thoughts.