one problem with an ever-accelerating earth that i haven't seen addressed, is the problem of time dilation. unless you believe in a steady-state infinitely old universe, you cannot accelerate forever. at relativistic speeds, time dilation becomes so severe that we would very quickly reach the end of the universe. depending on which hypothesis of "the end of everything" holds true, that is variously hypothesized to "only" be about 200 billion years. which would go by very quickly - after a year of acceleration, we would already be at relativistic speeds, and from our point of view, the cosmos would quickly burn out and die (or collapse).
which then begs the question, "relative to what?" in the scenario i specified, the earth is accelerating relative to the cosmos or some averaged "reality", which is what will someday collapse or cool to near absolute zero. but the fe argument would be, i would guess, that the entirety of existence is accelerating along with the earth. which begs other questions, but in and of itself at least "resolves" the problem of already having exceeded the lifetime of the universe since i was born.
i don't pretend to have a working intimacy with the math of str, so i am basing these arguments on my own extrapolations of the thought experiments einstein laid out, and others have done since. (e.g. the common illustration of being able to circumnavigate the known universe at near-light speed, several times in a single lifetime...however there would be no earth or possibly even milky way galaxy as we knew it to return to after just one trip around...)