I started out completely anti-screen and we lived that way for the first 2-3 years. We are still TV free (and I have no intention of changing that aspect!) and all of our viewing comes through DVDs from the library, DVDs that my sister records and sends over and the internet.
Two things have made me really back down on my anti screen stance.... the biggest one is the explosion of expression that my kids have got from having regular, exterior English input and the other is realizing that my son actually seems to really like acquiring information through watching. It's really cool listening to them fluidly using vocabulary and expressions that never would have come up in our day-to-day lives!
I talk with my kids about the fact that watching things is only one way to experience life and that it's important to experience a big variety of things in different ways. My son will come to me saying "I've played Legos, made a snack, played outside, been in the sandbox, played with the dogs, checked for eggs, been to homeschool group, etc and now can I watch something?" I would feel totally comfortable saying that I am willing to allow screen time in because it gives access to more of that variety.... in your case Hungarian). And I'm pretty comfortable with saying no when I feel like they've already been saturated.
I totally struggle with how much I feel comfortable with and judgements about quality. My kids are too little to be able to truly surf the net, so end up with a range I choose and I don't actively seek out Spanish/Catalan programming online (although if we lived in the US I would) because I feel that we already have more screen time than I would like. Ironically the library is my biggest struggle because I feel there is a lot of stuff I wouldn't choose to offer, and a lot of stuff not available in the original language (so we're stuck with Juan y Tolola instead of Charlie and Lola). However, I really want my kids to feel free to explore in the library and wouldn't police books, so I struggle with suddenly doing that with video.
Also, living in a hot climate, I notice that the screen at midday becomes a lot more attractive to all of us.... shades low, a cool drink while we snuggle on the sofa. But on the other hand, we're out cruising around the neighbourhood at 9pm, so it's more like the rhythm of our life changes when it's hot. As the weather changes I start to shift when we watch.
Another possibility to consider is bringing some other stuff in to where you're watching. My kids will set up elaborate Playmobil scenes and build forts while watching. I also tend to use that time as a chance to put nourishing snacks in front of them as they are actually still enough to eat!