@user1744397 What sentence on the relativistic rocket site do you mean? Do you mean the statement "Einstein postulated that any experiment done in a real gravitational field, provided that experiment has a fairly small spatial extent and doesn't take very long, will give a result indistinguishable from the same experiment done in an accelerating rocket." This could easily just be a somewhat sketchy true-enough-for-purposes-of-present-discussion statement aimed at laymen.

Note that the relativistic rocket page is part of the collaborative Usenet Physics FAQ, and elsewhere in this FAQ we find this page which deals more specifically with the equivalence principle, and which defines the equivalence principle to say the following:

"The metric of spacetime induces a Minkowski metric on the tangent spaces. In other words, to a first-order approximation, a small patch of spacetime looks like a small patch of Minkowski spacetime. Freely falling bodies follow geodesics." So, it contains exactly the same qualification about it only working to first order that I directed your attention to before.

What's more, it also says "once we subtract off the first-order effects by using a freely falling frame of reference, the remaining second-order effects betray the presence of a true gravitational field". In other words, even in an arbitrarily small patch of spacetime, there are second-order effects that would not be seen in flat SR spacetime.

If you decide to arbitrarily choose to trust a single statement in an online FAQ over a seemingly more technical and precise statement in the exact same FAQ, and ignore references to more mathematical statements in actual physics textbooks, I think you are cherry-picking whatever arguments will allow you to believe you've found a "flaw" in physics, rather than making an earnest attempt to determine what GR actually says.

As for "Do you think the predictions at the relativistic rocket site (e.g. 6.6 years travel time to Vega) are meaningless or invalid?"--no, I don't, but I do think you can't naively assume that they are justified by the equivalence principle, rather you have to first do some kind of GR based analysis to show that the large-scale curvature of spacetime between Earth and Vega is such that departures from SR in predictions about overall travel time will be small.

Presumably the FAQ author knows of such an analysis, and that's the basis for the statement that the SR approximation becomes significantly inaccurate for "For distances bigger than about a thousand million light years". The point is that this sort of analysis of where SR analyses break down has to be done on a case-by-case basis, and you have no basis for assuming the SR analysis will work to second order in the case of moving away from near an event horizon.

As for the cannon, I agree that your distance from the cannon must be increasing the moment after you're fired, but it's a non sequitur to say that this means your distance from the horizon is increasing. A "hovering" cannon is maintaining a constant distance from the horizon in schwarzschild coordinates, but this doesn't mean it's doing so in whatever approximation to an inertial coordinate system you are using (which again, will only act like an inertial one to first order).

Likewise I don't necessarily think it's correct that "If Earth was smaller but had the same mass, the initial recession speed would be higher. Initial recession speed would always be the speed needed to escape." You seem to be assuming that because the relative speed would increase in an inertial frame in Newtonian physics, you can use some kind of equivalence principle argument to generalize this to a local inertial frame in GR; for reasons I've already discussed, I don't think these sorts of

handwavey verbal arguments are trustworthy, an actual mathematical GR analysis would be needed to find out what happens in the chosen coordinate system, both in terms of the exact answer and the first-order approximation.