I like that you practiced on a old grill which is a good idea.
I am a retired aircraft worker and have drilled out thousands of rivets,of many different types. Raised head as used on the Weber vents countersink or flush to the surface rivets . I would do is get two pieces of metal that you can review together, with several pop rivets that are of the same type as used on the vent. Before you drill use back up support behind the revit so your drill dose not fall thru. As stated used a smaller drill bit to start then finish with a larger bit. Finding out the size needed is done by measuring the bottom side of the rivet diameter. Use a variable speed drill start at slower speed and raise your speed as you enter the rivet. To fast causes bit burn, also use a good quality bit cobalt is great stays sharp and is harder to burn.
Grinding works if you have a steady hand and strong carbide dices as they can Crack and explode sort of and scratch your lid, they can also bit into the rivet sort of grab, chatter and bounce off the rivet and scratch your lid.
What ever you choose patients and prep is key to a good result.
Then as stated above replace with quality stainless hardware.
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