Isn't that trivially fixed by raising court costs (that should go to whoever loses the suit) to cover the cost of judges, jury, admin expenses etc? I don't get the impression that this would make the justice system that much more prohibitively expensive than it already is, and would allow the legal system to scale to the case load

No, because the limitation is not money. More money does not magically make the humans in the profession be able to handle higher case-loads, nor magically produce new lawyers and judges. The bottleneck is time that each case takes to be properly and thoroughly adjudicated, and neither "more money" nor "more people" can accelerate that. While it's certainly correct to say that more staff could handle a larger number of cases, a. more staff = more cases, but more money doesn't speed up those cases, so there's still not really anything to be gained in terms of efficiency by increasing individual case costs. And b. if the solution was as simple as "hire more judges", it would have happened already.

Courts aren't lacking in budget to hire more people. They're lacking in people available to hire, with the specific expertise that they need to fill any gaps. The legal profession, at least in the US, consistently has some of the lowest unemployment rates across the board. Unlike over here in the tech sector, the scarcity is in available talent, rather than available jobs.