Evolutionary Tree

[Esquilax]

The study of fossils assumes everything biological is related and attempts to chart them accordingly.
The study of genetic mutation rates assumes everything is related and therefore rates of changes can be measured going back.
The study of parallel and convergent traits assumes (be definition) that a common ancestor did not have the traits being studied. 
The study of biological systems assumes everything is related so compares different systems in different species to find similarities to establish potential building blocks that could have evolved.

I could go on...but the point is that if common decent is disproved, there are ramifications in every sub-field--and many existing conclusions would have to be thrown out or at the very least, reexamined.

If you think any of those things are assumptions, instead of conclusions we've reached based on a wide breadth of observations, then I don't know what to tell you, since you're clearly ignorant of even basic history of science.

See, we have these things we do, called observations, that allow us to make deductions about the nature of the world around us. So when we see, say, that organisms closely related to one another tend to look like one another, that leads us to the conclusion that physiology is an indicator of relation. And when we discovered the genetic code and found that- why, would you look at that?- genetic similarities, which are expressed as physiology, tend to match with that previous deduction of ancestry, we have yet further confirmation of our initial deduction. And then we go further, we look at genetics from organisms we definitely know are related, and we confirm our thinking yet further because their genes have specific similarities and even identical pieces, so now we know that those things are suggestive of common ancestry. There's more, but the point is that, from these disparate points of knowledge, we can string together a definite pattern that allows us to conclude that common ancestry, probabilistically, fits the available data best.

You don't assume common ancestry for any of this, because you don't need to. Each individual "assumption" you see is, in actuality, buoyed up and supported by the relevant facts of that specific field, the understood biological mechanics that we've seen play out countless times that, while they add up to support of the common descent idea, are individually just... things that demonstrably happen in biology. So when you say that the study of biological systems "assumes" everything is related, you're questioning common ancestry, but in a more immediate sense you're questioning genetic testing, and what we know about shared genes in related organisms, up to and including simple paternity, which would be ludicrous.

If you have to question precepts of biology so well understood that they amount to legal proof in a court in order to make your point, well, your point would probably be better of not made.