I face this a lot and I will share how do I handle this one.
Do you confront them with a polite reminder of health issues, or that you don't appreciate them handling the food?
Of course, you should confront them because it's a matter of health issues. For those who use hands, I say like,
This dish appears to be tasty and you know other people will also like to take this. Therefore, it's not a good thing to pick this dish with hands. They have provided utensils for this.
You can skip bringing the topic of health issues unless it is necessary, because most people will definitely relate this with health. After saying this, I hand over them that utensil and they use given utensils after this for all dishes.
For those, who use wrong utensils, I say like (almost in the same way above but a bit different),
This dish looks good and using the wrong utensil might make this dish less tasty. So, please use this right utensils. Give them the right utensil
Now coming to another question,
Or is it best to ignore it, and just ask staff to provide clean utensils or a fresh "uncontaminated" offering?
I'd say don't ignore it and ask staff to provide clean utensils. You can also ask them to fresh offering and wait, but other people in the line might not wait. So, just ask them to provide fresh offering quickly in a polite way, like
A wrong utensil has been used for this dish, making me reluctant to take this dish. Please provide fresh offering.
You can try more direct approach if they don't listen to you at first.
I never had case of meat spoons being used in vegetarian dishes, since I am vegetarian and never attended parties where they serve both veg and non-veg dishes. Though you can say like,
This spoon is used in meat dishes and many people who are vegetarian wouldn't like it. This buffet is for everyone, so please mind other people too and use the right spoon.