To start with, I'll just say that I won't really address anything the posted video said (whether right or wrong) and just stick to strictly answering the question:
What would happen to Earth and life on Earth if all other heavenly bodies outside of our Sun, Moon and our planet Earth ceased to exist in some non-destructive way?
That being said, let's be clear about the parameters here, and what is happening in this scenario. I'm assuming that the universe, our solar system, and Earth all formed exactly as they already have, with the single exception that now some god-like being, simulation admin, or whatever controls out universe just removes everything in the universe except the Sun, the Earth, and the Moon. And by remove I mean just blink out of existence (against all logic and physics). Not blow up, not throw into a black hole, just remove. As with any question that takes the form "What happens to physics when I violate physics?", the answers can be a bit hard to specify. But, to the best of my ability, what would happen to the continued existence of Earth and life?
There are two main factors to consider I think. What we'd lose within our solar system and what we'd lose outside our solar system.
Inside Our Solar System
In this scenario, we're losing all the planets except Earth, all the comets and asteroids, and all the gas and dust out there that isn't strictly "the Sun". To be honest, this isn't a hugely impactful change because we're only losing ~0.13% of all the mass of the solar system. What changes internal to our solar system might we expect?
- The Sun's orbital motion would suddenly and almost completely cease. This would be almost entirely due to the disappearance of Jupiter. Aside from our Sun, Jupiter is the big heavyweight in our solar system and does actually have a measurable affect on the Sun's motion within our solar system, with respect to the "center" of the solar system. You'd see kind of a discontinuous kick that might be jarring to Earth, potentially adjusting the orbit.
- No more meteor showers. At the time this answer was written, Earth was at the height of the Perseid meteor shower, but all that debris would be gone so no more showers. For that matter no more Haley's comet or any other comets that one can sometimes see in the night sky.
- As a tongue-in-cheek observation, all Martian meteorites are also gone and visiting Antarctica and the UK National History Museum becomes slightly less fun.
- I'm assuming because all spacecraft and rovers derived from Earth they do not disappear, so now we've got a variety of rovers and spacecraft just drifting through space. Perseverance would be unlikely to tell us anything interesting though because I doubt the developers who wrote it handled the case where Mars blinked out of existence. While Perseverance can technically talk directly to Earth or else relay to nearby spacecraft, it's unlikely it could orient itself and point at a target long enough to say anything useful, so sorry Perseverance, looks like you're on your own.
Otherwise, the Moon would continue to orbit the Earth, Earth would continue orbiting the Sun (albeit, potentially in a slightly different orbit) which would continue to shine. Frankly I can't think of much more that would happen inside our Solar System due to losing the planets.
Outside Our Solar System
Here's where things get interesting. We're losing a lot of stuff outside our solar system - in fact most stuff. Your question wasn't very specific though. You simply specified "heavenly bodies". Depending on how that is interpreted, the answer could vary.
Heavenly Bodies == Objects with Mass
If I interpret "heavenly bodies" to just mean objects with mass, but not objects of pure energy, then we're losing all the stars in our galaxy, the rest of our galaxy, all nearby galaxies, and all the other galaxies out there. All black holes are gone. All interstellar and intergalactic gas and dust is gone. Suddenly the universe doesn't have Dark Matter. What we're keeping are things like all the photons out there, Dark Energy, Vacuum Energy, and the energy in the curvature of space-time.
This kind of has a somewhat boring answer under this scenario. Because it takes time for things like light and gravitational affects to get to Earth and because the Universe is just mind-bogglingly large, for most of the rest of humanity's existence, we don't really see much of a change at all. Even though all the stars are now gone, it'll take a very long time before we stop receiving light from them because most of the light is streaming towards us still and will be for many long years to go. Over time though, some of the nearest stars will blink out of the night sky as their last rays of light finally hit Earth.
A few notable examples include Alpha Centauri (4 years), Sirius (8 years) (the brightest star in the night sky is gone! The next brightest, Canopus, will be around for an additional 302 years before disappearing itself), and Procyon (11 years).
Otherwise, most everything you can see in the night sky will still be observable for most of the rest of human existence, even if the objects don't exist anymore. What's more, the gravitational effects will still be felt since gravitational changes travel at the speed of light. Our solar system will, for the most part, continue to orbit the center of our galaxy, even though the mass it's orbiting doesn't exist anymore! It'll take tens of thousands of years before our orbit will start to change (due to the gravitational disturbances of the missing mass finally catching up to us), but who knows if humanity will be around then.
What we may experience though are sudden jolts in our path through space as these blips in the gravitational fields pass over us - the most prominent coming from the disappearance of the SMBH at the center of our galaxy, although that won't occur for about 26 thousand years.
Outside of our galaxy, it'll be millions or billions of years before light from distance objects stops reaching us, so likely that'll never come to pass.
Finally, it's possible the universe will explode! With basically no mass, but still tons of dark energy (in this scenario) the expansion of the universe might be kicked into overdrive and very possibly the very atoms of our bodies will be ripped apart with that super-expansion! But there are a lot of if's behind this idea so who can really say if that will happen.
One interesting additional change would be that because the interstellar gas and dust has disappeared, our sun's atmosphere will have free reign to expand as much as it wants. Voyager I and II, with what little life they have left, will possibly notice that they've once again passed into the protective sheath of the Sun's magnetic field and solar wind (doubtful if it they the energy anymore to support such a measurement).
Heavenly Bodies == Objects with Mass-Energy
Now we're upping the game. I'm now interpretting "Heavenly Bodies" to include not just the objects described in the previous section, but all the other objects in the universe which have no mass, but have energy. So what does that add to the mix of disappearing objects? All the light pervading the universe is just gone (most of which is light radiated from the recombination era in the form of the Cosmic Microwave Background radiation). The energy built up into the curvature of the universe is gone (essentially meaning the universe stops being curved (or so I'm interpreting it)). And finally, Dark Energy is gone. This last one is a bit dubious as we don't actually know what Dark Energy is so it's hard to say what taking it away means.
This scenario is a bit more scary because the universe will be drastically different. Some changes I can think of off the top of my head.
- No more light at night from stars. During a new moon, night time will be very dark, almost unnaturally dark. Although, away from city lights, you'd have a much better chance of seeing the Gegenschein which would be pretty cool.
- The static of the universe, the CMB, would be gone. That'd be unfortunate since we can learn a hugely large amount of information about the nature of the universe from this light.
- Because I'm assuming dark energy goes away now, the expansion of the universe would likely stop (assuming our ideas about dark energy are correct). This really won't have any noticeable affect on our day-to-day back on Earth. Just something to note.
- The universe might still just explode. It's hard to say what may happen when you take all the energy out of the curvature of the universe. It may spring back like a rubber band and destroy everything (which would just be the Earth-Sun-Moon system at that point).
Overall I'm sure there are more effects one could dream up, but this list is getting pretty extensive already.