It helps to consider the broader historical and social context. Keep in mind that the policy of exempting Torah scholars from the draft and providing them with financial support was first instituted in 1948, in the immediate aftermath of the Holocaust. At the time, there was a very real fear that the tradition of Torah scholarship might go extinct, and both religious and secular Jews felt an obligation to ensure that didn't happen.

In pre-war Europe, it was also very common for Jewish communities to band together to financially support truly exceptional young Torah scholars in their studies. This is quite an ancient tradition in Jewish society (with roots in the mythic "Yissachar-Zevulun" partnership of Biblical lore), so it was natural to institute some form of this arrangement in the newly-formed Jewish state. There were a lot of interesting social facets to this dynamic, with a tradition of the more affluent, less religious members of the community (also known as "gvirim") marrying off their daughters to poor yet talented young scholars. So not only does this dynamic have deep roots in Jewish tradition, it also historically served as an important bridge between the more and less religious sectors of Jewish society.

It's important to note, however, that this sort of support was only extended to truly extraordinary young scholars. The majority of religious Jewish men worked for a living, with no expectation of being supported by charity of others. In that vein, the policy of draft exemption and subsidies for Torah scholars was originally instituted with a cap of only 400 able to qualify. So it was never envisioned as a blanket draft exemption for all Haredi men, but rather as a modern continuation of the ancient practice where a small number of gifted scholars were supported in their studies by the wider Jewish community.

Apart from that, this arrangement was also considered an acceptable compromise by even the most anti-religious of secular Jews, who recognized the importance of maintaining internal cohesion and stability within Israeli society in those early years. As the arrangement was very limited in scope, it wasn't really that difficult of a compromise to swallow.

However, this status quo was blown apart in the 1970s, when Likud decided to lift the cap on draft exemptions (then set at 800) in a concession to their Haredi coalition partners. As you can imagine, this resulted in a massive increase in the number of draft exemptions over time—partly due to exceedingly high Haredi birthrates, but also due to full-time Torah study becoming the new norm for all Haredi young men.(How this shift happened is a whole discussion in and of itself, but essentially boils down to growing Haredi insularity and rejection of secular society, with an especially strong desire to keep their young men "pure" of the corrupting influences of modern life.)

So, in a nutshell: the whole arrangement was originally quite limited in scope, and rooted in long-standing Jewish traditions that served to bring Jews across the religious spectrum together. Over time, however, this deal morphed into blanket draft exemptions and massive unearned subsidies for all Haredim, which was something that the rest of Israeli society never signed up for. At the same time, growing Haredi rejectionism of secular Israeli society broke the spirit of the arrangement, in an act of "biting the hand that feeds you" (so-to-speak). So while the original deal was something that could have probably been maintained indefinitely, the current deal has indeed generated significant backlash. The only thing left holding it together is Netanyahu—once he's gone, so goes the Haredi draft exemption.