What would happen if every single defendant insisted on going to trial rather than plead guilty?

Something like that in Canada. You are also guaranteed a trial in “reasonable” time. Our Supreme court in a case a while ago determined that should be 18 months for simple cases, and up to 30 months for more serious crimes (if I remember the numbers right). Since our provinces were cheap bastards and skimped on the court system, this decision resulted in a large number of pending cases being thrown out. Now, prosecutors have to consider the timeline. You still hear from time to time a case taking far too long to get to trial and being tossed for not being a “speedy trial”.

I like the system I am told the British use - the prosecution can only offer a 2/3 reduction in the sentence for a plea deal. This forces them to prosecute for a reasonable sentence - the theory being, if you are guilty and know you will lose, 66% of the sentence is better than 100%. If you think you are innocent and maybe you can win, you aren’t gambling like some American cases, 10 years vs. 1 year. (Or in the case of Aaron Schwarz, 35 years vs. 6 months - so he committed suicide.)

The OP forgot another threat - I have only been notified I was up for jury duty once. Similarly, my wife had the same notice. In both cases, the jury summons was cancelled before it got to trial. The defense threatens “if you take us to trial, we’ll ask for a jury trial” which is a lot more hassle and more unpredictable. Canada has a limit, IIRC the charges have to carry 5 years or more to have a jury. In the USA, everyone threatening to need a jury pool - creating a need for 20 times as many jurors as normal to handle all the cases - could really bog down the system.