As far as Haredi parties go, Shas always, sice its formation, participated fully in governance by taking ministerial positions UTJ refrained form taking ministerial positions, claiming an inherent problem with being a partner in things the government might do that go against the Torah. They could be members of the coalition in the Knesset but, they felt, by not taking ministerial positions they were not really members of the government and did not bear any shared responsibility for decisions taken that might involve chilul shabbos or other violations of halacha. Eventually they started taking deputy minister positions, still claiming lack of shared responsibility as long as it is not  a full miniterial position (and sometimes they even basically ran the ministry with a puppet minister from another party holding the ministerial position on behalf of UTJ. Eventually the Agudat Israel faction of UTJ also started taking full ministerial positions while Degel continued to only take deputy minister roles at most.

And now, according to news reports yesterday, Degel is considering taking full ministerial roles just like Shas and Aguda. I have not seen any explanation as to what changed in their policy to consider this other than their ongoing competition with the Aguda faction as this would allow them to demand the top position for their faction in the next government and not let Aguda have it. Degel's rabbis might or might nto approve the change, but thew that they are considering it is noteworthy enough.

Personally I support this. Their lack of shared responsibility is a farce and just something to make themselves feel better. Fuller participation makes less of a joke of the government and more direct involvement on their side.

That being said, in recent weeks I have heard the argument (raised when talking about the draft law) they like to raise of "ten li yavneh vchachameha" - give me Yavneh and its sages - like the famous request of Rav Yochanan Ben Zakkai of the Roman general Vespasian where he said just leave us alone to learn torah and we wont bother anyone. In Israel it is used as well regarding the draft law - just leave us alone and let us learn torah and we wont bother anyone, ten li yavneh vechachameha, just leave us alone dont bothe rus and we wont bother you. You cannot really say ten li yavneh vechachameha when you are involved in running the government and deciding how the country should function and how people's lives are affected. A true Ten Li Yavneh Vechachameha argument would work if they pulled out of politics and said just leave us alone to learn torah, dont bother us and we wont bother you. When they are increasingly involved in governance, the argument ten li yavneh regarding one aspect of life in Israel when there is involvement in everything else is a false narrative.