Of course you have to throw the man out at first if you want to get him out. Fundamental baseball says that you always try to get the lead runner out if you can. If the shortstop felt the man on second would be out at third, he would have gladly given up the easy out at first base. Fail in that and now you have runners at third and first. So he wisely let the man take third and took the easy out at first.
From the runner's side of things, if the man on second didn't think he could make it to third, he wouldn't have taken off for third, he'd have stayed where he was.
From the batter's side of things, if you can't hit safely, you want to advance the baserunners. That happened here. The batter was out but he advance a man one base.
From the official scorer's side of things, this could have been recorded as a sacrifice if he felt that the batter intentionally gave himself up to advance the runner.