Can you do that, just see how it goes and have the injection etc if you need it?
Yup. My hospital's policy is to inject with synto regardless. However I asked if I could try without (and I had heard that the synto shot works quickly & efficiently if PPH begins to occur) and they allowed me to do so. (I had a birth plan on file but that midwife hadn't read it. I had prepped DH beforehand to tell the MW what I wanted.)
Placenta came out whole within 5 minutes. Unfortunately I did have some heavy bleeding and they asked me if they could give me the shot because they were worried. I ended up needing two shots to control my bleeding, but otherwise all was fine and I'm glad I gave it a try.
As for delayed clamping, I believe that's becoming a more popular choice and my MW was certainly very happy to wait for the cord to stop pulsing.
GL!