The edges of that pond, look higher than the surrounding ground, except for the bit of the ground to the left (in the picture).

Usually, no one builds a pond like that, UNLESS it is a groundwater pond. (Or they have another source of water, like a water well.)

I have two groundwater ponds on my farm that are probably 90 years old. There is a large ring of giant cottonwood trees at the same elevation all of the way around my largest pond. There are zero mature cottonwood trees below that level. The current groundwater level at our farm is approximately 9 feet below the water level from 90 years ago. This is almost certainly due to significant withdrawals of water from the aquifer for agricultural use.

There is a chance that you had a nice groundwater pond, a long time ago when it was constructed. However, IF it was a groundwater pond, AND the level of your aquifer has substantially dropped, then you may be fighting a losing battle. It may be very difficult to seal a pond that was not designed to be sealed in its original design.

[All of that above is just speculation from a guy on the internet that cannot see your ground!]

OTOH, if there is a dam that washed out due to heavy flow of surface waters, then that suggests my speculation above is probably wrong. In that case, one weak spot where leaks might be likely is "welding" the core trench in the dam repair to the old core trench from the original dam to make a water-tight seal. I believe that is very difficult to accomplish without proper compaction equipment.

Finally, a pond liner may be an affordable option for a small pond of that size. Pricier than some repair options, but a MUCH higher success rate. You kind of implied you do have some other source of water to fill the pond? If so, that helps with the liner option. If not, hopefully some experts will drop in with some good "pond leak repair" advice!

Good luck getting your pond eventually full!