I'd probably TIG it starting along the seatstay weld, and continuing up through the toptube.

It is quite possible that the crack does extend through the front part of the seat tube which will be hidden. You might see it by inspecting the inside of the seat tube. It would be hard to repair. Perhaps that is one that could be drill stopped. It may be only marginally structural, but could increase the risk of recurrence.

Otherwise, I wouldn't drill it out of you're planning to TIG. The end of the crack will disappear with the welding process.

I generally don't grind welds, and so the repair will be visible.

As mentioned, with good weld penetration, you'll need to ream the seat tube.

How tall is your seatpost? Good fit? I'd make sure the post extends down at least a couple of inches beyond this junction.