A.J.
I used the "model politeness, always remind them to show manners, and immediately remove for discipline if they refuse" approach, and I've got three polite kids, including my 2 1/2 year old. Yes, sometimes one of them will need prompting (your daughter won't do it on her own all the time at 2-or even 4!), but they know refusing is not allowed, or fits or anything else. They were "forced" to react appropriately until it became their own habit.
When I couldn't get instant compliance with them being polite to someone in that moment in public or whatever, I would only ask once, and discipline as soon as I could after the fact with calm explanation as to why the discipline was occurring. I would also say to the person they were rude to, "I am so sorry she is being rude, she knows better than that". This was a huge red flag to my kids who KNEW some follow up was imminent when I had to apologize for them, and it was a good preface to the explanation and consequence after the fact to clarify it. In addition to the direct consequences of thier decision to be rude, at other times, we would discuss respect and manners and they were always praised for good ones.
Yes, there will be the rare embarrassing moment when your attempt at getting them to be polite is ignored and you feel bad, but 99% of the time they will do the right thing if the alternative is not OK with them. You need a firm, effective deterrent, or you'll have one of those bratty kids who is rude to people. They're pretty common! Good work getting serious about this. Alone time isn't very unpleasant though, she may not care enough to stop that behavior if that's her only consequence.
I think it's better to prompt them and enforce, then to "give up" and not make them claiming "it's not genuine anyway if I make them" like some people do. IMO that's not confronting the situation. No kids voluntarily do it. Good manners are taught and enforced.