If you had hired a competent lawyer then the lawyer could have forced the company to produce the contract. (And forced the judge to play by the rules.)
The problem you have is that you thought you could use logical arguments to win. You can, but you have to structure those logical arguments through a specific legal system.
Think of a basic logical argument. A > B. B > C. Therefore, A > C. Simple. Easy. Logical.
Except if you want to make the argument, you have to know how to prove A > B. And prove B > C.
If you don't know the mechanisms the law allows to make these proofs, then the logical of the argument is valid, but your proof is unsound because you can't back up the underlying assumptions.
A lawyer should be trained to know how to get evidence into the record to do this. (Some forget their training, though, and some never got it.) Further, a good trial lawyer will know the best ways to present the evidence to the fact finder (judge or jury) so that not only will the evidence support the logical assumptions, but the evidence will support it in the strongest possible manner.
Another way to look at it is that you simply don't know the structural rules. It's like showing up to a baseball game expecting it to be played by football rules. The underlying logic of how you win—score more than the other guys—is valid, but the ways you get to that result are different.
Finally, just because small claims court is easier for non-lawyers (by reducing the amount and types of procedural rules), that doesn't mean the law changes. You still have requirements that must be met to sustain a legal case against another party. If it was really important, you should have sought legal advice.
As for lawyers not taking your case—that I can sympathize against. I can say I've taken small claims cases for cheap, in part to help folks out and less to make money, but many lawyers won't. This is for two reasons: (1) the money isn't much and (2) many lawyers feel less comfortable in small claims than in higher courts (because there are different rules).