As of patch 6.0 Fish eyes now applies to all Stormblood fish, except for Legendary fish (so, Ruby Dragon, Stethacanthus, Drepansis, Xenacanthus, Warden of the Seven Hues, and Unconditional)
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So reading this I don't think the issue is (so far) an issue with fishing. It's an issue with how you're fishing.
As Brightamethyst said, you should be using Patience II and Makeshift bait while Mooch II is on cooldown. Not doing so is hooking yourself in the foot and you're making it harder on yourself.
Shark Tuna is up rather consistently compared to other fish, the layers of RNG for shark tuna are there - but are pretty much bog-standard fishing rng. You'll find plenty of other fish in a similar boat as shark tuna.
You can also increase your changes (affect the RNG) by using certain baits. For example, using a Versatile lure on fullmoon sardine instead of say - Spoon worm - you're going to see a lot of issues and are (again) putting yourself at a disadvantage.
The game does tell everyone these things - when Patience II is good, which baits are better in what environments, etc. There's very little (outside of window information) that the game DOESN'T tell you about fishing mechanics, or when to use certain skills. It's not well presented, or... people just don't read things like tooltips.
Fishing is equivalent to bluemage, but for DoL's. It was a limited job before limited jobs were a thing. Comparing it to and expecting it to be like MIN/BTN is just going to set yourself up for frustration. It's a minigame tied to a job.
Big fish don't need such a high chance to get away, they are already rare enough.
The bait system is stupid, outdated, and it clutters the inventory. Get rid of it.
Oh well. One less thing to do.
The right bait helps focus in on getting a specific fish and you want to eliminate that. Versatile Lure's equal chance to catch everything is where we end up with no bait to mess with. Think about what you're asking for before you go and ask for it. Maybe fishing is not for you.
Gosh, imagine defending having to carry over two bags' worth of bait.
Fun fact!
Versatile Lure will catch all fish in the game except for straight catch legendary fish (so like, Cinder Surprise, Aquamaton, Sea Butterfly, etc).
After ARR, there's a huge amount of bait that you can get rid of once you caught the fish, even more you can get rid of if you go the Versa route.
Bait bloat is mostly an ARR problem, and once you clear that out (or just ignore it and buy like, 20 at a time when you go fishing cause it's cheap) - bait management is significantly easier.