This is something that has worked for me. Yes, I do get about 1-3 silver saps per tree. I'm completely talking out of my ass, but just try it.
take a tree and cut it in whatever fashion that breaks the logs asap if not instantly. Then, run/jump in one direction for more than 50 blocks. Run back and check the drops. Do NOT touch the leaves ever. Wait for 5 seconds, run in another direction for just over 50 blocks. Come back and check the drops. Do this 4 times. Then finally Change your video settings from fast to detailed(I can't remember what its called atm) and back again. Run in another direction come back and check drops.
I do not have the absolute reason why this works. I know that there is supposed to be no difference in hand breaking leaves, sickle breaking them, or letting them naturally decompose. That being said, I have never received a silver sap manually breaking the leaves, but the previous method described has always worked. Standing right around the cut silver wood waiting for a sap has never worked.
Maybe there is something that plays with the rolls in vanilla mechanics, or the rolls are refreshed when you leave the immediate vacinity and come back (drops are not viewable until with 20 blocks or something), I don't know. I'm just saying I have no problem saying I can have a surplus of silverwood saps if I wanted to.
And for TC4, just 3 days ago I lumberjacked a silverwood, did my method and got 3 silver saps. Did it earlier before that and got 1.