How can I fix this table?

If it's still solid and you still like it I'd just st sand and stain. Cutting board or piece of tile next to grill would be all I'd add to it. 

Character. I agree. But my OCD wife would not approve. Not at all!

Here's a quick and easy way to deal with the burn mark and give you a nice surface for hot things to avoid future burns.

For a table I used to have, I wanted a granite surface to set down very hot items (like plate setter, etc.). So I went to a local place that installs granite countertops to look around. They had a whole lot of sample pieces, roughly 12x18 inches in size, for customers to ogle. I asked if they had any more scrap like that, and the sales person said no, but they would gladly sell me those samples because they could replace them easily.

I bought three of those 12x18 pieces for $25 total. Put one under the egg (with Egg feet to raise it for air space), and two in different places on the table top. Worked beautifully.

If that siding is wood I would move it away. My house fire ignited the siding and it went up like a rocket

A couple ideas:

1) stain the whole top dark to match
2) sand it out 
3) replace the entire top
      A ) wood
      B ) stone/concrete     
      C ) SS
4) take a torch and burn the entire top to match. 

I sanded my down this year and restrained and added a couple of coats of poly. I attached a before and after. 

Thanks @epcotisbest A note to the OP the dents, dings and burns I had made neat character marks with the new finish.