It sounds like you are probably a very experienced "trial attorney," which is different from a "pretrial litigator." It's important to understand that litigation in small firms is usually very different from litigation in large firms.
In large firms, many of the cases are very big and can go on for years before they either go to trial or finally settle. As such, most of the litigation consists of pretrial activities (discovery, expert witness reports, motion writing/briefing, hearings, etc.) that lead up to a possible trial. In certain large firms, there are junior partners who have never been to trial and some junior partners who have never argued in court. In many mega firms, some associates rarely (if ever) take depositions or argue in court. (Often, high-profile and high-paying clients want the partners to handle these aspects.)
On the other hand, these associates develop other skills-they are adept and skilled researchers, they have a uniquely focused attention to detail, they have fantastic Bluebooking and writing skills, they can write motions and eloquent briefs that are nothing short of works of art, and they can effectively organize and produce millions of documents in large-scale document productions. While they may rarely step into court and argue motions, their fantastic attention to detail and ability to navigate through the pretrial process are what high-paying clients want.
On the other hand, in a very small firm that has an active trial practice, the skills of a typical associate are much different. Usually, the cases are much smaller and go to trial much more quickly. Therefore, the litigators must be good trial attorneys. In many small firms, litigation associates attend court for the first time shortly after they begin working. Many first- and second-year trial attorneys are comfortable speaking to judges in court, while this would scare many associates in larger firms. Trial attorneys in small firms are often more adept at thinking on their feet while being questioned by judges in court. They often possess much greater ability to apply the rules of evidence. As a result of their day-in and day-out courtroom experience, they can cross-examine witnesses very effectively and connect with juries in ways that many litigation associates in large firms cannot.
So when you say that you "run circles around [your] friends who are litigators in the big firms," I can certainly understand and appreciate this. However, keep in mind that you are comparing apples to oranges. You are able to run circles around your friends, but only in regard to "trial litigation." When it comes to the type of pretrial litigation that is inherent in large law firms (which is not your area of expertise), your friends can probably run circles around you in certain respects. Sure, they may not be able to argue in court, but they can probably teach you a thing or two about running a large-scale document review that involves several million documents and a host of complex privilege issues. No one type of litigation is better or worse-they're just very, very different.
The best litigation practices are those that have both pretrial litigators and trial attorneys, since the skills are quite different. However, in most litigation departments in large firms, a majority of the litigators are pretrial litigators since pretrial is such a dominant portion of the practice.
The reasons you may not be getting interest from large firms at your level are 1) they are credential-focused (yes, grades still matter) and 2) they strongly prefer mid-level litigators who have immediately transferable skills that can often only be gained at competitor firms with similar practices. Because much of your experience is as a trial attorney in a very small criminal firm, you don't have experience practicing in a large competitor firm.
Also, more importantly, you have focused on handling criminal-defense matters, but these types of "street-crime" cases are rarely handled by large firms, which generally focus on commercial litigation. (An exception is white-collar defense, which is often practiced in larger firms since the defendants are usually very wealthy and can afford the large-firm rates.) As such, there would likely be a substantial learning curve involved were you to join a large firm.
Thus, rather than retrain a trial lawyer from a small firm, a larger firm will often prefer to hire somebody from its competitor across the street (somebody who is more experienced in handling commercial disputes). Indeed, the ramp-up time would be much shorter for such an attorney than it would be for somebody who has focused predominantly on criminal-defense matters.
This is not to say in any way that criminal defense is somehow not as "good" as commercial litigation. (In fact, many people would argue that it's much more exciting and fun.) The point is that it's just a very different ball of wax, and firms prefer to hire people who already know their ball of wax.
You did not mention your reason for seeking a position in a large firm. While I have a hunch it's probably due to the recent raises in salaries, I don't want to make this assumption. (Plus, this issue will open up a whole new can of worms best saved for another Q&A.)
However, I'll say this: since you really enjoy being in court, you will probably be much happier and more fulfilled in a trial-focused practice. Most of these practices are at smaller or mid-sized firms where the billing rates are lower and the stakes are not as high, but the cases are more likely to go to trial. Best of all, since you enjoy being in court, you'll have a lot more fun. With the number of unhappy attorneys out there, the importance of having fun and enjoying your career is something you should never lose sight of while looking for a new job. Good luck!
See the following article to learn how much the law school you attended matters when moving to a new firm:
How Much Does the Law School You Went to Matter When You Lateral Firms?
Please see this article to find out if litigation is right for you: Why Most Attorneys Have No Business Being Litigators: Fifteen Reasons Why You Should Not Be a Litigator