That is a beautiful pond. Good news is it looks to be mostly grass, and the slope doesn't look terrible.
If you keep looking, you'll probably come across a used walk behind trimmer for $75-$150. Buy it, use it, resell it and you're probably not out much if you don't like it. If it has just been used as a trimmer and not as a substitute bush hog, it probably will have less hours than a regular mower.
This is the brushcutter I bought.
ECHO SRM-265U Emission-Compliant Powerful Brushcutter - ECHO USA (5 year consumer warranty)
I used it with a metal saw tooth cutting head to clear out a patch of kudzu and briars. If you put a metal grass blade on it, it would slice right through your tall grass. It won't grind it up like the walk behind string trimmer, but it will cut it down. That's all you need to clear a path.
They also make a metal blade conversion kit that HD carries for about $39 that you can use to adapt their other trimmers to a metal head. It includes a blade. So you could buy one of their $220 Echo trimmers and convert it if you wanted. It just so happened that the closest Echo dealer had the larger unit that he had ordered for a customer who didn't like it, and I was able to get it for a bit more.
I suspect that say 30 years ago, companies could build self propelled mowers with exposed blades that would really chop through grass without clogging up, and then products liability concerns forced them to start using enclosed decks. The enclosed decks require more HP, and the machines have gotten to be expensive.
The walk behind string trimmers have the exposed cutting head, but I suppose the use of trimmer line means they don't throw objects quite as far as an exposed steel blade.
I think either a walk behind or a shoulder trimmer will cut a path for you with some work on your part.