Why does logic emphasize tautologies rather than contradictions?
I think the answer is something like the following:
We can't cover all cases by contradictions. For example: if I have to prove that p implies q, then I can prove it by contradiction, not(q) implies (not)p. By proving that, we basically proved that whenever p is true, q is true. We have no information about all other cases.
I think there's an even simpler reason: it's hard to preserve truth when all you start out with are falsehoods!
I take it that you've been studying axiomatic systems of logic (I'll restrict my focus to propositional logic). Typically these systems will have a number of axioms (which will be tautologies) and an inference rule or two--- typically something like modus ponens.
The concept of a valid inference is one that preserves designated values. In classical propositional logic the designated value is "truth", and so a valid inference is one that preserves truth. How do you prove this? Well you start off by assuming you have true propositions to reason from and then show that repeated applications of modus ponens will never take you from a truth to a falsity.
If you start with only contradictions then you will never get from a truth to a falsity, but only because all of your "axioms" were false to begin with! Such a logic might make falsity its designated value and gerrymander a notion of valid inference where a valid inference is one that preserves falsehood. But the standard notion of validity will do no work in the setting you've described (assuming I understand correctly that you want to replace all tautological axioms with contradictory ones).
Also, depending on what inference rules you have, your system could get messy very quickly. If you still allow ex falso quodlibet (literally "out of a falsehood anything (you) please", sometimes called the rule of "explosion": from a contradiction anything follows) style inferences then since you have let in contradictions as axioms you will have every single sentence expressible in your language as a theorem of your system--- surely an undesired result.
So, why don't we base our logic on contradictions? Because contradictions are bad!
(NOTE: Paraconsistent logics like Graham Priest's "Logic of Paradox" typically invalidate ex falso quodlibet. In Priest's case it is because he is a "dialetheist" and believes there are true contradictions. Removing the explosion rule is a way to handle "true contradictions" (which, itself, reads as a contradiction to me) without having them "infect" your entire system and trivialize inference.)
In the main line of the Western tradition of logic (which follows the Aristotle) contradictions are emphasised by being the limit of what can be said. They are not the centre of a logic or given attention for themselves. Having said that Chrysippus, the Stoic logician wrote 23 books (chapters) on the liars paradox, and in fact it was this paradox that inspired Godel and his theorems and hence played a large part in reinvigorating the moribund field of logic.
The Catuskoti in the Eastern tradition shows that contradictions were more central to their thought.
In propositional calculus or truth-function logic there are two ways to show validity. One can either take a well-formed formula and show that it is a tautology using a truth table or one can use a proof system and show by derivation that given the premises, if any, the conclusion results. That is, one has two approaches, a semantic approach to validity using the true-false meaning of propositions and a syntactic approach to validity using a derivation.
To make sure these two approaches are clear first consider the truth table for this single well-formed formula which is the tautology '(P ∧ (P → Q)) → Q'.
Note that 'T' appears on all rows in the last column after the header. This truth table shows that the well-formed formula is a tautology.
There is also a way to show the validity using a proof system. Here the premises are 'P' and 'P → Q' and the conclusion is 'Q':
A basic question is do these two methods give the same results? Or as forall x (page 144) puts it:
If you can show that 'A' is a tautology using truth tables, can you also always show that it is true using a derivation? Is the reverse true? Are these things also true for tautologies and pairs of equivalent sentences? As it turns out, the answer to all these questions and many more like them is yes.
There are two basic results, soundness and completeness, which forall x describes in Chapter 20.
Now consider the question:
Why does logic emphasize tautologies rather than contradictions?
At least one of the reasons to emphasize tautologies rather than contradictions is that tautologies pair up with the derivations of proof systems.
References
Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/
Stanford's Truth Table Tool: http://web.stanford.edu/class/cs103/tools/truth-table-tool/