There are two different "circle around distant bodies" here that are probably causing some confusion. Classically, if near your appliance you can see stars and

There are two different "circle around distant bodies" here that are probably causing some confusion. Classically, if near your appliance you can see stars and Mars through the window on a cloudless night, a flash of light from your appliance can in principle be seen (e.g. by a really good telescope orbiting one of those bodies). Indeed, some of the light from a laser pointer shining out your window will go to infinity, and has some astronomical chance of being stuck the photon ring around a black hole. It works just like you being able to see through your window starlight or (in principle) a bright flash on the surface of Mars. However, this is not the sort of "goes around Jupiter" that we are talking about; we are talking about something like the probability of one microscopic component of your appliance's flash of light scattering off or being absorbed by a microscopic component of said window (for example, in part of the spectrum in which the window is not transparent), where calculating that probability considers an infinity of possible paths. Indeed, an infinity of the path-integral trajectories of one such photon (that classically is definitely intercepted by the window) will go out to infinity, an infinity of those trajectories will go out to Jupiter, an infinity of them will go to somewhere very near the window pane, an infinity of them will speed up and slow down along the way rather than travel at the speed of light at all times, etc. We then want to assign a probability to each of these paths. The idea is that trajectories that are not like a classical path between appliance and window are suppressed because (somewhat technically) they [a] have the same magnitude of probability amplitude, but [b] have different phases, [c] different changes in trajectory have different changes in phase, and [d] for "longer" paths compared to the classical one, a small change of trajectory makes a big change in the action compared to Planck's constant, so [e] for sets of small changes in "longer" paths the phases change rapidly and thus are highly likely to cancel out. So most of those infinities are effectively irrelevant and we can focus on the smaller infinities that make a marginal-to-relevant change to the total probability amplitude and figure out a way to calculate it from them (e.g. by sampling).