Thanks again for your answers.
I'll try the streaming from my pc even if I'm a bit afraid it will lag.
And I guess it's mandatory to have a pc running in the house? Because I have a laptop so it's not cool to have to turn it on every time I want to watch TV. If I knew that before, I would just have bought an HDMI cable to simply plug my pc to my TV and watch what I want...
I don't see the interest in the Apple TV if we still need a pc running.
I bought it because I wanted to separate these two things...
It is probably the wrong device if you want "to separate those 2 things." It can work pretty well without a PC running but only if the videos you want to watch are sourced from Apple (purchased/rented content). Your first post is ambiguous if your videos are purchased from iTunes/Apple or if they are perhaps videos you've made yourself or DVD/Blu Ray rips, etc. If the latter, you probably do need a computer running iTunes in the house to host & serve them to
TV (unless you go with an app solution that can connect to network storage, but either way, you will need some computer-like hardware running elsewhere in the house on which to store the videos).
The point of owning an
TV in that scenario is that it makes it very easy for you and others in your household to play those videos. You/they don't have to attach the laptop to the computer each time you want to watch a video. Nor do you have to be interacting with the laptop to watch videos either.
TV works in the classic way (interface with a remote from the couch). If your laptop is a Mac, you also have the easy ability to "airplay" video on your laptop screen to the bigger TV screen wirelessly. If you are NOT single, living alone and that laptop goes out of the home with you,
TV is still back at home to serve others in the household when they want to watch or do something with it.
TV still has plenty of capabilities when the laptop is NOT at home. Your own personal videos not purchased from Apple will typically not be accessible but others at home can still buy/rent movies from Apple, Podcasts, Apple Music, run many Apps, etc. I suppose the best way to think about the computer connections links back to your original want- to sync videos onto
TV. While it can't do that (and doesn't have very much storage for many videos anyway), think about the computer dependency as being the storage for up to any sized personal movie connection. That storage is simply OUTSIDE the box, on hard drives you can connect to a computer.
Since a laptop is a mobile
device and assuming the videos you want to be able to easily watch are not purchased from Apple, consider picking up any used/refurb Windows PC or Mac and dedicating it to this purpose (being an always-on iTunes-running computer in your home). Or perhaps you have an old laptop that still works but was retired because the new one is so much faster/better, etc? The iTunes-running computer does not have to be a very powerful computer- just one good enough to run iTunes.
As to "afraid it will lag", it generally will not unless you have poor wifi inside your home (which has nothing to do with the speed of your Internet connection). If you have this problem, you probably need to upgrade/reposition your wifi hardware. Or, if you can connect a Ethernet cable via home network, that will be plenty fast for these purposes. Remember there is a big difference between Internet connection & speed and home network speed. If there is lag for videos stored on your own laptop, one doesn't even need an internet connection at all to watch those (so lag issues are probably in quality of home network technology or perhaps videos not formatted well for
If the above doesn't do the job, there are other
TV-like boxes that are basically big video-storage hard drives in a box. Your TV may even have a USB port to which you can attach a hard drive loaded up with video files to play right through the TV's interface.
Or, you can adopt other system options via apps that can work with network attached storage for the videos (instead of a dedicated computer running iTunes, you have another piece of hardware- a NAS drive- on which the videos are stored). As mentioned earlier, PLEX is very popular and it can work like this. Instead of a computer running iTunes, some options like PLEX can access a video library stored on a NAS.
If you are pretty deeply invested in the Apple ecosystem, you'll likely find it easier to dedicate a computer running iTunes instead of using something like PLEX. PLEX is great for what it is but it is not as readily connected to all things Apple, so it may become a matter of using PLEX at times and then hopping into native
TV at other times. While that may seem rudimentary, it is pretty nice being able to get the full
TV experience and all benefits (including Apple DRM-protected access) from within a single UI (vs. hopping app-to-app). If there are others in your home that may not be as tech savvy, it also is typically easier for them to work within a single UI instead of hopping into another app(s) for some video.