Certainly looks like an amateur hour install! Yes def shove the frame back against the surround again to its original position. But for it to stay there you probably need to fix it again.
If it’s moving freely then the glass won’t crack but if you did not want to run risk of breaking the pane you could remove the double glazing panel by carefully prising away the plastic trim that holds it in and then get her all in position and replace pane afterwards.
To fix it and remove the drafts see if there is an easy win to put a few screws through side of frame into something solid( you would need window pane to be out for that) and then find a way to carefully squirt some expanding foam behind it and it will expand and hold everything together. Then use frame sealant round outside for a pro finish.
Some old school builders got uppity about using expanding foam for window installs as it was seen as ‘cheating’. But it only took one call back from a customer over a cold draft coming in to have them all converted and swearing by the product.