First, some definitions.
The limits of observability in our Universe are actually set by cosmological horizons, of which the "observable Universe" is just one type. These depend on not only whether can we detect the thing from Earth, but also on when was the signal produced?
There are various types of cosmic horizons:
The particle horizon sets a limit on the distance that can be seen due to the finite age of the universe - this is likely what you meant by "observable Universe." That is, the particle horizon represents the largest comoving distance from which light could have reached the observer by a specific time. Due to the expansion of the space of the Universe, this is not the age of the universe times the speed of light, as in the Hubble horizon (see below), but rather the speed of light multiplied by the conformal time.
The Hubble horizon is a theoretical horizon defining the boundary between particles that are moving slower or faster than the speed of light relative to an observer at a given time. This does not mean the particle is unobservable, since the light from the past is able to reach the observer. In the current models of the expanding Universe, light emitted from the Hubble horizon would reach us in finite time. (these details depend on the sign of the Hubble parameter).
The cosmic event horizon is the largest comoving distance from which light emitted now can reach an observer in the future. The current distance to our cosmic event horizon is about 16 billion light-years, which is within the "observable" sphere given by the particle horizon.
There are also "practical horizons" such as the surface of last scattering for photons (ie. recombination, origin of the cosmic microwave background), and suspected surfaces of last scattering for early universe neutrinos and gravitational waves.
Lastly, depending on the expansion of the Universe and whether we continues expanding forever, there could be a "future horizon," which sets a limit on the farthest distance that can possibly be measured in units of today's proper distance.
Keep in mind that this terminology is not always used rigidly and consistently, for example:
"Sometimes astrophysicists distinguish between the visible universe, which includes only signals emitted since recombination (when hydrogen atoms were formed from protons and electrons and photons were emitted)—and the observable universe, which includes signals since the beginning of the cosmological expansion (the Big Bang in traditional physical cosmology, the end of the inflationary epoch in modern cosmology)."
is it really "nonsense to even talk about" objects currently outside the observable universe but may become visible in the future not having any gravitational influence on us because gravity travels at the speed of light, or is this a useful concept even if perhaps an inexact expression when GR is embraced?
I hope with the definitions above, it is clearly NOT nonsense and clearly depends on what we mean by "observable Universe." Even taking the simplest example of the particle horizon, the Copernican principle implies that every point in the Universe has its own particle horizon, which implies that the objects near the edge of our particle horizon could be effected by objects beyond that edge. We could indirectly observe those effects after sufficient cosmic expansion. One can imagine more complicated scenarios, but I think this makes the point.