If aged 2 or 3 just let them get on with it, moving them out of the way if necessary!
By 4 or older they tend to be a little bit steerable but emotional or completely batshit, so I would definitely take away from the area to a quiet place and that would either put the brakes on it, or we'd talk a bit, remind of behaviour expectations and that would work but when in batshit territory, did have to sit on floor and restrain a few times!
But yes I agree it does sound a bit much to expect, you do have to plan the day around them a little bit when they are young. Can you go more often for short trips, order things online, or get somebody to babysit (or at least a friend to help distract!) if you need to do a long shopping trip? If you have a few bits to get then you can break it up a little by going to shops they like to play in etc or popping into the bank to use the bead toy things etc, bringing some snacks to eat on a bench, whatever it is. 3 hours straight in a shopping centre I couldn't have done when DS was little. 30 minutes is about right, possibly 40 with bribery of we'll go to the library afterwards or we'll get a milkshake or we'll go to the pet shop to look at the fish, whatever it was (doesn't have to be expensive or cost money).
FWIW I don't see a problem with doing tickle fingers in public. It definitely does help if you can let them wander a bit as well, obviously not running into unsafe places or touching a load of things in shops, but following their lead and letting them explore is okay and can help them be more amenable. Lots of warning in advance of what is going to happen (When we go into X shop we have to hold hands. Can you see it? When we go through that door there.) (When we get to the road, you need to either hold hands or sit in the pushchair. Which would you like?) Also, give her little jobs to help. "I'm looking for a blue t-shirt. Can you help me find a blue one?" and if you like you can pretend to be silly so that she can "help" you. "I need something to put on my feet but I can't remember what it is. Is it a potato? Is it a hat? OH!! That's right, it's socks, silly mummy, how did I forget. Now what kind of socks shall I buy..." DS used to love trying on all of the hats in H&M. Also praise when she is helpful/walking/holding hands nicely. It can be a bit exhausting, but, again, another reason why 3 hours is ambitious. I definitely agree with taking a buggy if you're going to be out for a while as well so she can have a rest and so that you have a way of containing her if she is getting uncontrollable.
Good luck!