It's too bad the search function for the Wattage List sucks so bad (especially since it's a Google product). We went through all of this 15 years ago so if you could search (which you can't) you'd find your exact argument from around 2003 or so.
So, to address some of your points:
1. The idea of TSS was based on TRIMPS, so the odd formulation is a heuristic form based on TRIMPS' heuristic. The simplified analytical form (IF^2*hours) is easier to calculate but harder to explain.
2. You (and many others) have been puzzled that TSS isn't additively linear. There are lots of things that aren't (for example, variance: if you took the variance of 1 hour of constant power at FTP you'd get zero, if you took the variance of another hour of constant power at 50% of FTP, you'd get zero -- but if you concatenate the two into one long ride, the variance *isn't zero!*). Likewise, if you multiply everything in that sequence by a constant, the variance doesn't change by the same constant! These may seem like odd examples, but the variance is the p-norm where p=2; and as you point out, NP is the p-norm where p=4.
3. It's sort of useful, especially given what we had available at the time, but it wasn't then nor is it now magic. And, it's sort of not useful, if you expect a single number to really tell you everything you need to know about how a ride fits into the bigger picture of training. There are lots of different measures of central tendency, including means, medians, modes, weighted means, truncated measures, etc. We don't think (at least, we shouldn't) that any single one is "right" and tells us everything. In fact, we often look at several measurements of central tendency and evaluate them in unison and in context. That's a long way of saying that TSS can be useful, but not always, and sometimes you have to consider it in the context of other measures of training load.
So, to address some of your points:
1. The idea of TSS was based on TRIMPS, so the odd formulation is a heuristic form based on TRIMPS' heuristic. The simplified analytical form (IF^2*hours) is easier to calculate but harder to explain.
2. You (and many others) have been puzzled that TSS isn't additively linear. There are lots of things that aren't (for example, variance: if you took the variance of 1 hour of constant power at FTP you'd get zero, if you took the variance of another hour of constant power at 50% of FTP, you'd get zero -- but if you concatenate the two into one long ride, the variance *isn't zero!*). Likewise, if you multiply everything in that sequence by a constant, the variance doesn't change by the same constant! These may seem like odd examples, but the variance is the p-norm where p=2; and as you point out, NP is the p-norm where p=4.
3. It's sort of useful, especially given what we had available at the time, but it wasn't then nor is it now magic. And, it's sort of not useful, if you expect a single number to really tell you everything you need to know about how a ride fits into the bigger picture of training. There are lots of different measures of central tendency, including means, medians, modes, weighted means, truncated measures, etc. We don't think (at least, we shouldn't) that any single one is "right" and tells us everything. In fact, we often look at several measurements of central tendency and evaluate them in unison and in context. That's a long way of saying that TSS can be useful, but not always, and sometimes you have to consider it in the context of other measures of training load.