Wow! Nice find. I just spend $16 on a 6lb container of Door county cherries.
I always use their cherries for lambics, but that's my personal preference. The beers you're tasting that taste like cough syrup are most likely made with cherry flavoring because it's cheaper. There's a reason why it's cheaper though, and I'm glad you've noticed that.
Real cherries won't give you that effect.
You could try a basic pale ale, or a wheat. Really though, lambics aren't that hard to make, I think people just overthink it and make it seem like they're super complicated.
Make a basic wheat beer, use some aged hops, add wyeast lambic blend and add cherries after 3-6 months. Age for a year total and your done.
Now, this is an ultra simplistic way of describing it. As you get more into sour beers, there's better ways to do things. But, this will turn out a basic lambic that will be more than drinkable if you allow enough time. Decoction mashes and longer boils are nice, along with the use of multiple yeasts and blending, but it's not absolutely neccissary.