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...I have never transitioned to tubeless tires on a bike (although I've run tubeless tires on all the cars for a long time now).

For those of you who do set up your bikes with tubeless, isn't it kind of a sticky mess to find the offending object that flatted your tire, to the point where you need to run a tube to get home, then insert a tube, and pop the bead back on, so you can inflate your new tube ? Or am I just imagining this ? The tire has a lining of sticky sealant goop, right ? Isn't that difficult to deal with when your leak on a high pressure road tire doesn't seal on its own ? Or am I just imagining a problem that does not exist ?