So because of an overly-broad warrant, or lack of training, or vague suspicions or even the mere chance that your data happened to be stored on the same box as Criminal A, we should just allow the FBI to violate the Constitution?

Reductio ad absurdum with your argument and we should just let the FBI run all internet infrastructure, because it is expedient for the FBI, it makes life easier on the FBI, and they can inspect your data to their heart's content. It is a much preferable situation to the one where an FBI agent has to do something complicated.

Nowhere do I see you advocating for the innocent whose rights are infringed upon.

The Constitution specifically states "unreasonable search and seizures". I did not say it should be easy for the FBI to collect. I specifically was referring to "reasonable". Again, if you were tasked with collecting this data, how do you do it reasonably? Sometimes the most reasonable thing is to simply copy the hard drives the data is living on.

The reasonable here does not mean what's techincally easiest or reasonable for the government, but what a reasonable violation of the suspects' or innocent people's privacy is, in light of the suspected crimes.

Grabbing the servers is easy, but arguably because so much unrelated data could be on a shared server these days, it's unreasonable to just grab everything. Many other people are affected, and unrelated data is collected and evaluated later for purposes unrelated to the original collection purpose. This circumvents a reasonable expectation of privacy, and strongly shifts the balance of power from the regular citizens to the agency hording the data. Especially in light of the plea-bargaining justice system in the USA, where the data can be and is used to threaten and coerce, this is worrying.

EDIT: missing word

Right, technical details aren't the real issue. Ultimately, the govt has to earn back the trust it has destroyed, or face rebellion. That's how trust works.