Does "to err on the side of ... " indicate wrongdoing?

In the phrase “to err on the side of…”, the meaning of err is closer to that of the Old French errer, which is “to stray”. The image implied is that, if you can't for sure walk on the right line, you might want to err on one particular side rather than the other.

So, just to make sure I answer your actual question: no, “to err on the side of…” does not indicate wrongdoing.

"To err on the side of caution" means that if you are wrong you will at least be wrong in a cautious way. Let's say you have some chicken in your fridge. You're not sure if it's edible or not because you don't recall how long it's been there. Even though you are hungry and want to eat it, you throw it out to avoid any chance of food poisoning. Then your wife/husband/roommate gets home and says "I just bought that chicken today!". Well, you erred in throwing it out, but it was through caution.

Basically it means to act conservatively even if your actions are more conservative than they need to be.

No, it just means that in a situation involving an unclear distinction between what may be right and what may be wrong, the speaker is going to try to choose a course that is closer to the right side.

It's like when surgeons cut out tumors. They cut out healthy tissue as well to leave clean "margins" — erring on the side of clean, in that case.

No, the phrase is used to describe where the margin of error is placed. (Although, that might not be the exact origin of the expression.)

To err on the side of caution is to choose to be as cautious as to be sure to always be cautious enough, so when uncertain, rather be too much cautious than too little.

It has to do with increasing the likelihood of success or safety.

Merriam-Webster gives the following as an example: "We chose to err on the side of caution [=to be very cautious] when planning our investments." I despise this usage because it reduces "err" to "being on the side of", which is redundant. Why not just say, "we chose to be cautious"?

That's why I always preface "err on the side of" with "I'd rather" or something similar. Doing so gives "err" a reason to be there. "I'd rather err on the side of _______" means "making a mistake with this would not be as bad as making a mistake with that."