Due Process is for defendants:
The "fairness" provisions in Amendments 4-8 do not "typically" apply to defendants. They explicitly apply only to defendants. For instance, the 5th Amendment says:
No person shall...be deprived of life, liberty, or property, without due process of law...
The basic idea is that the public is best served by having a fair, impartial trials under law. As Paul Johnson notes in his excellent comment, the general presumption is that the state has much more power than defendants, and hence is more likely to pervert justice for its benefit. The "fairness" provisions aim to reduce the power imbalance by requiring the state to follow certain procedures, such as getting warrants, and so on.
Fairness and prosecutors
It is bad when either side perverts justice. That is why even prosecutors have some ability to challenge trial procedures that seem biased in favor of defendants. Thus, prosecutors cannot generally ask for a change of venue, but they can object to particular jurors. The idea, again, is that given its resources, allowing the state to move a trial would hurt defendants.
As Paul Johnson notes, the analogy of impeachment to regular trials breaks down at this point. The Constitution imposes only a few bare-bones requirements on the impeachment process. This lack of structure, when combined with the Constitutional provision that each House can make its own rules, makes it easier for House and Senate to do things that would not be allowed in regular trials, such as delay bringing charges (which would violate the "speedy trial" clause), or seat admittedly "not-fair" jurors.
What is fair?
The obvious response to charges that the process is "unfair," is to ask, "Unfair by what standard?" After, all the rules of impeachment are what they are. It is hard to see how using these rules to one's advantage is unfair.
It is important to remember, that as in regular trials, both sides are busy trying to put their spin on what is going on. If either side believes they will get more support by talking about "basic fairness," that is how they will talk. But this spin should not hide the reality that the impeachment rules are what they are. As with any rules, whether some tricky manuever within the rules is fair depends on whose ox is being gored. After all, most Republicans were thrilled when Mitch McConnell invoked the Constitution to justify delaying a vote on Merrick Garland, President Obama's nominee to the Supreme Court. Democrats were not. Now when delay does not serve Republicans, they are much less impressed with the Constitutional arguments.
