I guess it depends on what you mean by renounced. There are a ton of atheist Jews in Israel, including more recent immigrants, not to mention a ton more dual nationals. Obviously if you are openly hostile to Jewry itself or a public critic of Israel as a permanent Jewish state they probably arent going to let you immigrate, but I dont think there is any requirement that you accept the religion of Judaism so long as you dont belong to another religion.

On the issue of 1/4 Jews immigrating there. I have a feeling they are kind of selective on this one. If a half black person who is 1/4 Jewish tried to immigrate to Israel they would probably find excuses to prevent that, and I remember reading an article some time ago that alleges they do exactly that.

Judaism has traditionally been inherently obsessed with bloodlines because of the whole chosen people thing, and more recently because many ashkenazi Jews believe themselves to be of superior intelligence. The entire reason for the maternal requirement is that previous to DNA testing there could always be doubt about who the actual father is. The maternal requirement ensured they have indisputable blood ties to Judaism. As time went on Jews in Europe realized there are advantages to selectively letting some goy elites marry into the tribe, so standards were relaxed, but the obsession with kindship or bloodlines still remains, and in an area of DNA testing half Jewish is half Jewish and it no longer matters if the Jewish parent is a mother and father, aside from some orthodox Jews who want to follow the religion to the letter(most Jews today are secular).

It's also likely that half or 1/4 Jews are looked down upon by many other Jews. I remember some based half Jews on the reddit sub that said as much and would talk about the neurosis, hostility and manipulativeness of their Jewish family towards their non Jewish half.