There are two main problems with ranking the principles.

Firstly, it mostly doesn't solve the problem that you state, because unless Principle A is so much higher priority than Principle B that B is literally irrelevant, you will still have cases where there is tradeoff between a small amount of A and a large amount of B. Again, the courts will have to decide. (And anyway, the A-vs-A conflicts will also be up to the courts.) In general, people have an intuitive understanding of the ranking, an intuitive understanding that, weirdly, is a lot more concordant than their stated principles (or most people's ability to write down a particular value); and anyway, if a court somewhere has an objectionable weighting, all this means is that the legislature has to make explicit something about that particular choice of weighting (or the weighting in-that-situation), not everything. Also, if you do have the ranking written down, keep in mind that you don't automatically get your favored ranking. Anything you dislike about others' politics now would almost surely be pushed down into any rank-ordering if we had to do it at this point. So the change to full ranking would maybe be neutral, but almost certainly not a plus.

Secondly, as society changes, the imprecision in law actually allows flexibility for the system to adapt to a contemporary understanding without the massive burden of having to re-legislate everything. This is, on balance, an advantage, I think, as long as the institution of the court is kept sufficiently pristine. The problem has been not that the process is insufficiently democratic--good heavens, look at what Florida has been trying to do!--but rather that it's been too greatly swayed by political jockeying as opposed to a broadly neutral view of society. A court system protected from undue influence of politics can, as understanding evolves, use broadly-stated principles in the new context to shape the rules of society to be more workable for people therein. Otherwise it wouldn't just be women in Vermont who need permission from their husbands to get false teeth.

You overstate the paradox of intolerance with statements like this one: "we tend to be very wary of expressing intolerance of another person’s religion or culture. And this fear is codified in our legal system". But the degree of appropriate "fear" is not demonstrated to be a sufficient damper on intolerance of intolerance to cause any problems. Indeed, you seem to draw the boundary between tolerance and intolerance in a very weird place, as if you think "tolerance" means "support" and "intolerance" means "dissent": "it should be seen as perfectly appropriate (and in fact, encouraged) for us to express our intolerance merely by engaging in a civil discussion in which we explain our opposing viewpoint."

But this kind of action is precisely what Popper, for instance, advocated for instead of being intolerant! You can discuss all you want, and for as long as they'll listen, if you don't actually do anything, you are, in fact, tolerating their beliefs. Just tolerating. Not endorsing, not agreeing to disagree, just tolerating. "I can put up with it if they insist on believing that garbage (at least if they don't act on it)": that's tolerance.

So where most people overdo the paradox of intolerance by overdoing how vigorously you have to be intolerant of actual intolerance (i.e. how much do you need to fight fire with fire, or can you fight it with something less dangerous), you overdo it by making "intolerance" such a mild concept that you can't even tell someone that they're mistaken without being "intolerant". But because, as you agree, that kind of extreme "tolerance" is no virtue, the paradox does not apply because it doesn't matter if it's lost. The paradox still applies, though, once you place the tolerance/intolerance boundary far enough down so that "intolerance" is a really perverse and harmful thing.