The last time I looked at those rope repair kits quite some years ago, they were clearly marked for temporary use only.
The intent of such repair kits was to get you to the nearest place of service where the tyre can either be repaired correctly or replaced. The correct repair process is (as VC mentioned) to first remove the tyre from the rim and then inspect the internals for associated damage. I fall looks ok and the puncture was through the tread and not near the sidewall/tread junction, the puncture is cleaned up and the inside surface roughed up so a much room plug can be inserted into the hole from the inside and pulled hard from the outside so the mushroom top sits flat against the inside of the tyre. I believe they also use some vulcanising glue under the mushroom head and stem.
Such mushroom plug repairs and only on certain parts of the tyre tread are safe. In all other cases one should replace the tyre...
As for that tyre sealant goop, it’s also for temporary use only. The last time I looked at some, there was a clear warning not to use on motorbikes as it can impact tyre balance which is critical when you only have two wheels. It also had a warning not to use it on wheels with tyre pressure monitoring systems installed (though I think they have better fluids these days as Holden uses such in their tight arse standard tyre repair kits). ALL tyre shops hate sealant goop as it’s messy to deal and supposedly a smells though I’ve not had the privilege
I’d hate cleaning such crap from rims ifI worked in tyre repair business (which means it probably isn’t cleaned up terribly well impacting balance
The reality is, a quality full size spare tyre is the only safe* way to repair a flat for anything other than an emergency get me to a tyre joint situation. And don’t bag the guys if they hate working with wheels that have Tyre goop used on them or they charge more in such cases...
* changing tyres or using sealing rope/goop on the side of the road carries its own risks if one doesn’t have their mind on safety. The safety aspects of such roadside repairs should be taught as part of ones road skills development and such should be a part of any license test... but meh, our rulers probably think Darwinism