You are correct that many stock finishing tapes have relatively weak adhesive and are mainly decorative. In my experience, if you wrap the bars tightly and you maintain roughly even tension on the tape while wrapping them, the tape should not unravel. (Also, if the tape does unravel, it can be possible to unwrap it carefully and redo the job, assuming the tape is a decent quality.)

In the past, I had preferred to use electrical tape to secure the inner end, then wrap the finishing tape over the electrical tape. The suggestion to use self-amalgamating tape is a good one; I recently came across this tape and it does work. "Self-fusing silicone tape" tape may be a synonym. Either phrase currently returns a large number of results on the US Amazon.com store. I don't agree that you should necessarily use the provided finishing tape on the inside, then wrap that in self-fusing tape. I'd rather use that type of tape as the base or sole layer of tape.

I will note that wrapping your handlebars is an acquired skill, and many cyclists might only face this process once a year or less, depending on how much they ride. The opportunities to practice are less. It is definitely worth thinking about asking your local bike shop to do it, or to invest the time to buy one or two sacrificial rolls of tape to practice on your own. Also, in my experience, thicker bar tapes can be challenging to wrap. Currently, I can do a good enough job with a standard thickness tape, although I lack the skill to do the finishing cut cleanly. Thick tapes tend to defeat me (e.g. the 3+mm Lizard Skins tape or similar; these tend to be oriented at gravel riding for vibration damping).