Just my 2 pennorth for what its worth (which is not a lot) Like others, I have plugged the back tyre of the Blackbird and done several thousands of miles without an issue including several laps of the Nurburgring and Brands. If it is done properly with an internal mushroom and providing it complied with certain criteria then it is not a problem. The issue is with the external or roadside plugs intended to get you home. In these cases, they cause the tyre to delaminate and can actually cause an instant deflation which you do not normally get with tubeless tyres. I have delat with several crashes which occured directly as a result of the expernal plugs, but in nearly 40 years of crash investigation, I have never dealt with a crash caused by an internal vulcanised repair. If it meets the criteria, internal plug for me every time, and members of the BSi tyre committee (that I also used to sit on) agree.
I got a puncture on a fairly worn tyre last weekend on my FJR. Fairly worn is actually stretching it a bit - it was completely shagged by the time I stopped, think I'd been running at ever reducing pressure for some time which overheated tyre and accelerated wear massively. Fitted a mushroom plug which blew after 100 miles or so (still 70 miles from home). Went through 6 plugs in total to get home (slowly). On a tyre with plenty of wall thickness I'd be happy to use a mushroom plug but on a really worn tyre with little wall thickness it's a bit of a hit or miss. PS.. The tyre I was running was worn but still well legal when I left home. Planned on a 400ish mile run and tyre if it hadn't punctured would still have been way off wear indicators at journeys end.
The proper repair mushroom plug is fitted from the inside of the tyre. These should not be mistaken for the small temporary repair mushrooms found in puncture repair kits.
I would not hesitate repairing that tyre looking at the location. Can't tell the size of the hole though! I previous rear was had a puncture repair and I did more than 4k on it with no problem at all. In terms of high cornering speed, there was no difference. However if I were riding most of the time at 160mph plus, then I would probably consider replacing it in case the internal steel belt is cut at the puncture location. If the repair is done correctly I would have no worries. The only important point is the size of the puncture and the angle on the perforation. I would treat a double mushroom repaired with caution and watch my speed. And if the mushroom is not inserted at the same angle as the puncture hole, you may get a minute leak but if you have deep pockets and don't like your existing tyre, then it is a good excuse to replace