As stated on the trope's main page the main reason why the Conveniently Close Planet pops up and fails in Real Life is the humongous size of space. Simply put, distances up there are so vast that are beyond our comprehension as we do not find them in our daily life and, needless to say too, if a Star Trek-like show had to spend weeks just going from planet to planet (and that just within a given solar system, never mind interstellar travel) things would be more boringnote .
In the Analysis page of the Asteroid Thicket trope there's a scale-down model of the Solar System to show the real emptiness of the asteroid belt where the Sun ended up as a sphere with a diameter of almost 15 meters. Let's expand this model to the entire Solar System and we'll find that, to cite just some planets, Mercury would be orbiting at roughly 600 meters of the Sun, Venus would lie at 1.1 kilometers of the daystar, Earth at a bit more than 1.5 kilometers away, Mars at around 2.2 kilometers, Jupiter at 8 kilometers, Saturn at almost 15 kilometers, Uranus at roughly 30 kilometers, and finally Neptune at almost 47 kilometersnote . The sizes of all those planets at scale?. Well, Mercury would have a diameter of just 5 centimeters (ie: the size of a mandarin orange), Venus would have the size of a compact disc (a diameter of 12 centimeters), Earth would be 13 centimeters across (a bit larger than a compact disc), Mars would be a larger mandarin orange than Mercury (7 centimeters across), Jupiter a big beach ball with a size of almost 1.5 meters Saturn a smaller beach ball with a diameter of 1.2 meters (the main part of its rings would have a diameter close to 3 meters), and finally Uranus and Neptune would be smaller beach balls, with diameters of around 50 centimeters. Consider the distances between them given before and you can understand why we see other planets without optical aid as starsnote and how an alien ship that was transversing the Solar System at random would be extremely lucky to pass close to a planet or even the Sun (barring aside the possibility of the Daystar attracting it)note .
Just as planetary ring systems are the closest thing in Real Life to an Asteroid Thicket, the moons of giant planets are the closest thing to this trope the Universe offers us at least in the Solar System. For example, among The Moons of Jupiter Io and Europa can approach up to 249,000 kilometrs apart and in The Moons of Saturn Enceladus and Mimas can be so close as 53,000 kilometers in both cases well withing range of our technology in what refers to manned spacecraftnote and close enough that the Mark I eyeball would see from one of them the other as a body large enough to show phases and its most important surface features, just as happens with the The Moon as seen from Earth. Nonetheless they'd be much smaller than those drawings in comic books and the like where a moon fills a considerable amount of skynote
Outside the Solar System some planetary systems are far more compact than ours and you can find on them several planets, including worlds larger than Earth and up to Jupiter-sized ones, crammed within the radius of Mercury's orbit and even much lessnote . Such is the case of the recently discovered TRAPPIST-1 where seven Earth-sized worlds orbit a small star within a radius of 10 million kilometers (1/6th the radius of Mercury's orbit); if you were in one of the planets of that system, you'd be able to see the closest ones as distinct worlds showing phases -those that orbited the star closer than yours- and details. However, as the Solar System examples above, they'd be small (more or less as large as we see the Moon from our planet). Said worlds in astronomical scales, not in Earth ones, are quite close one to each other and an hypothetical civilization that appeared in one would have relatively easy to expand across themnote . But even those systems are quite distant of the Conveniently Close Planet trope; TRAPPIST-1 at the scale we're using would be seven roughly compact disc-sized spheres separated one from the other by roughly 15 meters orbiting a beach ball, and a ship that came in a random trajectory would be fortunate to pass very close to one of those celestial bodies.
One can but hope that in the day we finally leave this planet and interplanetary travel becomes as common as was before the era of jet planes to travel by sea between, say, New York and Lisbon and with similar travel times note writers will have more sense of scale and this trope will be a Dead Horse.