Your trees on the berm as so old, that some of the root systems may have died and decomposed.

It is difficult to destroy small water pathways like that with compaction alone.

Next time you plan on renting a small tractor for other work on the farm, you might consider draining and drying the pond a few weeks ahead of time. When you get the tractor on site, put the shanks down on a rented box blade (cheap) and rip the sides of your pond in a few directions.

Drive safely, and stay out of the bottom of the pond or any mucky places that will get you stuck. It doesn't sound as if the bottom of your pond is leaking, so just work the sides. IF the leaks are due to tree root invasion, then your leaks are most likely to be in the band from your seasonal high-water mark to just below your seasonal low-water mark. That is the portion you need to rip effectively. (This would be the time to spread any of the additives you choose - as described in the links above from jpsdad.)

After you have wiped out any of the existing water flow pathways, set your shanks back in the "up" position and use your box blade to contour your pond slopes back to where you want them.

Then lift your box blade and drive the tractor over your side slopes for some compaction. (Any weights you can safely add to the box blade and into the FEL would help for this step.)

A tractor provides a little compaction, but is designed to not get stuck so your large rear tires actually lessen the compaction of the tractor. Therefore, your tractor compaction might not be sufficient to re-seal the pond.

You might try filling the pond after that step to determine if the tractor compaction was sufficient. If not, then drain the pond again and try a portable compactor. There is an optimal moisture content for compacting clay, so you need to compact when your side slopes have only partially dried. (You might give them a light spraying of water to re-wet the top prior to compacting.)

P.S. Your eyes on the ground are much more accurate than distant friendly people on the internet. You must temper any advice with your actual observations.

My advice above is terrible advice IF your pond is built in sandy, permeable soil where they placed and compacted clay blankets to seal the pond. I don't believe that is the case, but you need to determine the best facts on the ground and then determine if the advice matches.

P.P.S. The tractor method described above is not that expensive, but is not guaranteed. Make sure you contact TJ. His method might be more cost-effective. Alternatively, the tractor method might stop 90% of your leaks and a subsequent application of TJ's product might stop the final 10%. There is no "perfect" advice, you just have to make your best effort and take those resulting improvements. If that is not enough, then you have to try another best effort.

Good luck getting your pond sealed so you can subsequently create the pond environment that your family would like!