The standard is just preponderance of the evidence, and not knowing what, if any, defense the repair facility offered that kind of testimony can be enough to win. Witness testimony is, after all, admissible evidence and if not rebutted certainly can be enough to win. I've seen cases won with just the testimony of the plaintiff making the case that basically goes "my version of events is at least slightly more likely than the defendant's". That's all it takes to win.
Let's look at it by the numbers, and let's assume that the repair shop (manufacturer) offers zero testimony and zero other evidence:
There are three possibilities about who damaged the screen. First is the OP, but the OP will testify that he didn't damage the screen. Let's say that the judge believes the OP. So, the OP is out.
Now, there are two possibilities left. First, that the delivery company damaged it, and second that the repair shop damaged it. That means that it's 50/50 whether one or the other damaged it. The OP simply has supposition about who damaged it, so the OP cannot win.
Except, of course, if the trier of fact makes a truly bonehead call...which is possible, and allowed for in my <=.1% possibility given above.