Having a very picky eater who is stubborn to boot, I've found the best approach to be "This is what we have" and then just go on with dinner. I do try to make things that he'll like as part of the dinner, but DH and I can't eat exactly what A's currently liking allllll the time (I have my grilled cheese limit!).
We have the "Try..." rule, too. But it's really more of a request when you think about it. You can't force them to eat something, but you can have consequences like Mommy isn't making something else; no dessert; you can get down from the table if you're done; or if you're hungry later on, then you can have the rest of your dinner.
Like Shiv, we talk a lot about how it's good to try new things. Sometimes, we don't like the food, but sometimes we LOOOOOOOOVE it! Does it lead to A trying most of the new foods we give him? Nope! But, we choose to take the pressure off ourselves and just say, "Well, such-and-such is on your plate in case you'd like to try it." Beyond that, DH will try to negotiate with him, but that can backfire big time if A gets fixated on the "prize" food. Eh, we're still learning how to navigate these waters, too. Heck, I was thrilled last night when the goober finally tried mashed potatoes and a bite of green bean.

Oh, and don't worry about Finn asking for chocolate and candy all the time. A does that, too, even though the child hardly gets any

It's not a reflection on you; it's just they have good memories and KNOW that stuff is yummy to them. They keep asking for it because, well, they're 3yos, and that's what 3yos do

Just keep doing what you're doing! Staying patient, calm, and talking can help, but also don't get too frustrated and feel like you have to do the whole song and dance routine to get him to try something, either. You're doing the best you can, but he has his free will
ETA: I think it's important to recognize when they do try something new--at least, in the beginning. Especially, if it's something they tried and DIDN'T like. Let him know that, "You know, buddy, you tried that and it wasn't very tasty. But now you know." The pleasing taste of foods they do like, I think, reinforces the trying rule in and of themselves.