Tank tape for tire patching - Enough friction?
Friction between tyre and tube is almost irrelevant to the normal situation. There's only movement at that interface at the contact patch, and it's normally small enough that it's done by stretch in the tube rather than sliding. Tubes don't abrade away over time. But when you have big splits in the sidewall friction does play a small part, in slowing down the rate at which the tube buldges out the sidewall before bursting. You're more likely to have a problem caused by the sharp edge or end of a fold in the tape grinding through the tube and causing a puncture.

In your case, it sounds as though you're going to a lot of effort for a temporary solution. I'm not sure why - if it's just to get you to the bike shop to buy a new tyre then maybe, but I'd be walking rather than spending time with tape on a solution that may not work.

All you really need here is tape that's reasonably strong and at least slightly sticks to the inside of the tyre. I prefer a boot made from part of an old tyre, and have seen someone use an entire old tyre to reinforce one like yours. That is a fair bit of work, you have to cut both beads off the inner tyre, but it does work.