A Rimfire Steel Challenge rifle will have quite different characteristics from one built for a precision match such as CMP Rimfire Sporter, and again totally different from ABRA sporter class. You do not want a heavy barrel for Steel Challenge because it will slow you down. It isn't impossible to get one that's quite good at both speed and accuracy, though, if you stay with high-end parts. You also have to look carefully at the rule books for the games you want to compete in. For example, Rimfire Sporter does not allow fluted barrels, which eliminates Kidd Lightweight and Ultra Lightweight from consideration if you want to do that match. For precision shooting you want a rifle that has some weight, to damp the slight wobble your body will inevitably create, especially in the standing position. The target rifles used in Olympic competition weight about 14 lbs for that reason. Same with benchrest.
I would focus on Kidd components. Kidd makes the best-quality upgrade parts for 10/22s in the galaxy. Their proprietary design single-stage trigger is unequaled by any other maker - it's the one you will want for Steel Challenge. The Kidd lightweight barrels are extremely accurate.
Another approach is to make a disguised match rifle by having the OEM bolt and barrel worked over by Connecticut Precision Chambering (ct-precision.com). Randy re-chambers the barrel and fine-tunes the bolt, making an ordinary rifle shoot almost like a Kidd. Ruger OEM barrels are excellent, but the chamber is designed to a loose standard for reliability.
Bottom-line, an all-around rifle will be a compromise in some respects. Just another reason to have several 10/22s.