Sorry to see there’ve been some poorly pups 
Mine is 18weeks. I’ve been letting him off-lead since he was 12weeks, taking full advantage of that ‘puppy recall instinct’ they have. Recall command wise, I use a whistle, an Acme 210.5. It has a consistent tone and if you have to replace it every one sounds the same so your dog will recognise it. It carries much further and clearer than my voice too (especially in windy weather) so he can hear it better. Given that we’re 18 weeks now, his ‘stay close’ puppy instincts are fading and yet he still recalls very well, probably 19 out of 20 recalls are first time comebacks. I recall him and he is rewarded with a click and a treat (we are clicker training too). The slight problem is that it can be a bit fiddly having the clicker, the whistle and a treat (high value treats like tiny bits of minute steak work brilliantly for us).
As for other dogs or people causing a distraction/run risk, the trick is for you to spot them and get your puppies attention before he/she does. Easier said than done sometimes, I know. You can use a long line if you want but some people find them a pita because they get tangled up and/or mucky/caught on things. I’ve got one but have never used it.
When I first let my dog ‘off the lead’ I used to leave his lead on but just let it drop to the floor. That way I had a bit more of something to grab if I needed to (rough playing, running off, eating things he shouldn’t, meeting people he shouldn’t etc). It did get grubby dragging on the floor though! After about a week of consistently good recalls I disconnected the lead completely.
@Nojeansplease, mine still has a few problems reading the body language of ‘no thank you’ dogs but he’s getting there. He’s much better at ‘asking’ if they want to play first and then coming back to me sooner if they give off negative signals. The problem comes when an on lead ‘No thank you’ dog is being walked with a playful dog. He finds it difficult to keep spatial awareness and stay out of the ‘no thank your dog’s space when he’s playing with the playful dog and ends up getting growled/snapped at. He could do with learning a bit more etiquette in that arena.