I partly agree, partly disagree. In general, it's a bad idea to invade early, because even if your laners have priority and can theoretically help you, there's no guarantee that they will. This obviously varies by elo and champion choice. If your mid laner is a Katarina for example, you can't really expect them to assist you in an invade at level 2 they way a strong early laner like Jayce could. Knowing this, it is entirely the fault of the jungler if they invade poorly without accurately judging in advance what kind of assistance either side might or might not receive.
However, there ARE cases where it's entirely within reason to expect (though not outright demand) a laner help you out in an early invade given two conditions which I previously mentioned:
- They have lane priority.
- They are a strong early-game champion.
Early invades can be especially deadly in low elo because snowballing is much more potent if players don't know how to play from behind. If a weak jungler (read: weak player) gets set back in the early levels, it's almost impossible to come back into the game, and usually leads to them tilting and possibly afking. And if you're playing ranked seriously, why would you not use every tactic at your disposal to gain an advantage?
As to your champion examples, I don't see Eve winning many 1v1s in the early game against fighters like Udyr, Xin, Olaf, etc. Tank junglers maybe, but Eve is pretty weak pre-6 without items, so invades are risky.
Shaco on the other hand can pull off some nasty invades if you know what you're doing and path correctly. Most of the time you don't even have to fight, just smite their camp and Q away (and if you miss the smite, you can still just Q away without losing much). Shaco vs. Lee, contrary to popular belief, is actually a skill matchup rather than a strict counterpick. You can block his Q with your box, dodge it with your own Q, and cut his W healing with ignite (which basically all Shacos take over flash). It's highly unlikely that you will kill him early game simply because his mobility is stronger, but if he commits to a fight and doesn't smite your box, you win the all-in.
Long story short, it's dangerous to make blanket statements like "never invade early with X champ" because jungling requires you to consider so many different variables that you can really only judge these things on a game-by-game basis. The most you can do is offer general advice. It sounds to me like you just got stuck with a few bad junglers and generalized all of those situations to the game as a whole; an all-too-common mistake tbh.