Without the extra rules, it might be possible to create a new billag?
Not really. Governments need always need a law to justify their actions. Without clause 2 there's no legal justification for running their own media outlets and therefore no justification to spend money for such things. Creating another billag needs some additional laws on top of that, otherwise they would have to fund it over taxes.
The extra rules sound like they want to prevent that.
That's right. So either they don't know that without clause 2 that would be illegal anyways or they think that the polilticians will break the rules anyway, in which case adding another 4 rules isn't really promising imho. No matter how you turn it, these guys have very little knowledge about how the constitution that they want to change actually works.
If I could vote I'd be convinced by your arguments.☺
If you're convinced by my arguments, prepare to vote No until the end of your life. I haven't seen one single initiative that didn't consist of such amateurish clauses. Additionally if everyone votes No our politicians will say "This is a clear Yes to our public media, therefore we won't even discuss abolishing it". I have seen that pattern over and over again. Sometimes you'll have to bite the bullet and take the shit that smells the least because there's never going to come something clean.
Have there been proposals to lower the amount and just spend it on some subset of it?
I would say you either say yes to public media, define its goals very clearly and then spend as much as it necessary to reach that goal, or you say to to public media and search for alternatives to reach desirables goals. Saying, Yes we want public media because we need unbiasedhaha news and a media outlet that shapes our education and culturewtf seriously?, but we want that to cost no more than X million CHF is pretty non-sensical. It's like letting GSOA people vote about which fighters our military has to buy. Of course they're going to vote for the most shittiest jet they can find, so they can say "Our military isn't up to its task, better abolish it". That way we won't reach our goals but spend a lot of money nonetheless. Nobody will be happy with such solutions.
A different approach would be "Here are the goals and that's what it is going to cost us. Yes or No, for the whole package?". I'm still waiting for the day that this happens in Swiss politics. Usually politicians don't make clear cost forecasts, either because they are incapable of doing so or they don't want it to know so they can say "That's going to be very cheap" or "This will cost more than we can ever afford.". Cost forecasts don't win elections, ideologically underpinned rants consisting of "facts" that are so exaggerated that nobody could possibly come to different conclusions does.