Is there a mathematical basis for the idea that this interpretation of confidence intervals is incor

We need to distinguish between two claims here:
Population parameters cannot be random, only the data we obtain about them can be random.
Interpreting confidence intervals as containing a parameter with a certain probability is wrong.
The first is a sweeping statement that you correctly describe as frequentist philosophy (in some cases “dogma” would seem more appropriate) and that you don't need to subscribe to if you find a subjectivist interpretation of probabilities to be interesting, useful or perhaps even true. (I certainly find it at least useful and interesting.)
The second statement, however, is true. Confidence intervals are inherently frequentist animals; they're constructed with the goal that no matter what the value of the unknown parameter is, for any fixed value you have the same prescribed probability of constructing a confidence interval that contains the “true” value of the parameter. You can't construct them according to this frequentist prescription and then reinterpret them in a subjectivist way; that leads to a statement that's false not because it doesn't follow frequentist dogma but because it wasn't derived for a subjective probability. A Bayesian approach leads to a different interval, which is rightly given a different name, a credible interval.
An instructive example is afforded by the confidence intervals for the unknown rate parameter of a Poisson process with known background noise rate. In this case, there are values of the data for which it is certain that the confidence interval does not contain the “true” parameter. This is not an error in the construction of the intervals; they have to be constructed like that to allow them to be interpreted in a frequentist manner. Interpreting such a confidence interval in a subjectivist manner would result in nonsense. The Bayesian credible intervals, on the other hand, always have a certain probability of containing the “random” parameter.
I read a nice exposition of this example recently but I can't find it right now – I'll post it if I find it again, but for now I think this paper is also a useful introduction. (Example 11 on p. 20 is particularly amusing.)

The very technical definition of a 95% confidence interval implies that if we were to repeat the experiment 100 times, 95 of those times our confidence interval would have the true population mean.
So saying that a confidence interval is the probability that the true population mean lies in it does not seem intuitively wrong, but technically the probabilities might work out differently depending on how we state it.
The definition relies upon the true population mean being a constant and our intervals kinda guessing it, whereas the usual, technically incorrect, intuitive answer depends upon our confidence interval being constant and our true mean shifting around randomly.
I hope this sheds some light on this topic.