What type of extrabiblical tradition?

Since so many Roman Catholic and Eastern Orthodox evaluations of Protestantism involve criticizing sola scriptura without making much of a case for an alternative, we should consider what's at stake. Not only does rejecting sola scriptura not leave you with Catholicism and Orthodoxy as the only Christian options to choose from, but it doesn't even come close to doing so.

Let's say that somebody found Papias' extrabiblical traditions about premillennialism convincing and added those to scripture as his rule of faith. Would accepting such extrabiblical traditions give you Roman Catholicism? No. Would it give you Eastern Orthodoxy? No. Would it give you the rule of faith of any of the other groups outside of Protestantism that claim apostolic succession, for example? No. Rather, it would give you something different than sola scriptura, but vastly closer to Protestantism than to those alternatives.

We need to keep in mind that there's a large gap separating sola scriptura from something like the rule of faith of Roman Catholicism or that of Eastern Orthodoxy. You can reject the former (sola scriptura) while still being much closer to the former than the latter (the rule of faith of the groups mentioned). When people refer to the importance of sola scriptura, they often have alternatives like Catholicism and Orthodoxy in mind. It doesn't follow that there's so much at stake when alternatives to sola scriptura are considered more broadly. When a Catholic or Orthodox tries to cast doubt on sola scriptura in a way that would still leave you a long distance from those two alternatives to sola scriptura, that long distance is important to note. We should also note the shortness of the distance between sola scriptura and other alternatives. In my example involving Papias and premillennialism, much less is at stake than would be involved in something like a conversion to Catholicism or a conversion to Orthodoxy. Or think of how individuals like Irenaeus (Against Heresies, 1:10:1-2) and Tertullian (The Prescription Against Heretics, 13) defined the apostolic tradition of their day. It was vastly different than what Catholics and Orthodox are advocating in our day.

Often, it seems that those who suggest that there's a narrower range of alternatives to sola scriptura have certain assumptions in mind that they haven't articulated or justified, assumptions their opponents don't accept. If the narrower range of alternatives to sola scriptura depends on those assumptions, then framing the discussion around that narrower range of alternatives is only as good as those assumptions.