Die grinders spin 12,000 rpms plus, they don't make torque.. A drill spins 300-1800 rpms & torque is what makes it work... So no the die grinder isn't a good idea.... FWIW the welding a nut method does work great, I've done it a few times & I've done recessed bolts, I think yours is borderline to deep but if your good with a welder & don't have an angle drill I think I'd give that a shot.. If you weld on it getting a drill started straight is gonna be harder for sure... But if it works you'll be saved the PITA of drilling & ez-outting the bolt..

"The Armies of our ancestors were lucky, in that they were not trailed by a second army of pencil pushers."