I see what you're saying here: if the media prints lies from a government it's not the media lying, it's the government

If the government manages to fool the media, yeah. If the government says to the media "the truth is X, but we're going to pretend that it's Y, so you print Y, ok?" and then the media goes along with it, then you can blame the media. In many cases, the media isn't able to fact check the things the government tells them. But, relaying what the government is saying is still important. Similarly, even though the media can't independently fact check the numbers that the Gaza Health Ministry reports, it's still valuable to have those numbers released too.

If the media is lazy about their fact checking you can call them lazy, but you can't call them liars, because lying requires knowing the truth and intentionally saying something untrue.

Here's the thing: if a government lies all the fucking time and the media keeps printing what the government claims anyway, then that makes them complicit in spreading the government's lies.

If the government says "the truth is X" and then the media says "X is true" then sure, you're right. But, if the media says "the government said that the truth is X", then it's up to readers / viewers to understand that the media isn't endorsing what the government said as being true, the media is simply telling you what was said.

The media doesn't get to wash its hands of the things it prints just because it puts "Israel says" before the headline.

Why should it need to wash its hands? That is exactly what Israel said. Because Israel has a complete ban on reporters in Gaza, for example, there's no way to corroborate or refute what Israel said. It's newsworthy to repeat what Israel said, but you can't blame the media when someone reads that and assumes that the government is telling the truth. As you said yourself, the government lies all the time, so why would you assume that "the government said X happened" means that "X happened".