Is Edwin Jaynes correct about statistics? I've recently been reading Edwin Jaynes's book, Probabili

I can't speak for all statisticians, but here's my view of these two questions:
First, I think that (1) and (2) are mostly the same question, each is asking about the relevance of the Bayesian-Frequentist debate.
In general, I'd say the debate has shifted significantly, notably via input from Deborah Mayo and Andrew Gelman (among many others...but I read these two the most), away from philosophizing about the nature of probability as applied to statistics to the idea of how we judge/check our statistical models. For example, a Bayesian claiming a 95% probability (posterior) of their model being correct is really not saying much, since we can't place that 95% in an objective context. Why should 95% be reassuring? What does 95% mean in terms of observable outcomes?
If you say, "Well, I accept my model as valid when the posterior probability is over 95%", then does that imply 5% of your accepted models are wrong (or something to that effect)? If so, you are implicitly using a frequentist calibration of your model. If you deny the validity of the above statement, then what can you offer to support confidence in your 95%? Why can't we say its 60%? How would you show that I am wrong in a non-circular way?
However, I digress a bit. My main point is not to criticize subjective probability, but to say that it's become largely irrelevant to modern statistical practice. Here's why:
Regardless of your philosophy, you need to validate your models!
No one, no matter how good their reputation or beautifully coherent their logic should get a free pass. You need to show your model is a reasonably good approximation of reality. To do this, you're going to need to compare your output to data, and if your outputs are probabilities, you can't get away from calculating frequency distributions/histograms/density estimates.
As far as I'm concerned, the philosophical part of Bayesian/Frequentist debate is largely over, with frequentism winning out as the preferred method of calibration or checking of models. There simply aren't many alternatives to showing "hey, my model is pretty good!" without appealing to some sense of frequency estimates. Logical coherence alone is not a valid way to empirical truth.
However, I'd also say that Bayesian models are strongly on the ascendance. Modeling your parameters as "random effects" from some meta model basically means you are entertaining a richer space of models, with the added benefit that your estimates may have less variance and lower overall error (if your priors are not completely off the mark).
So, yes, I'd say Jayenes was a product of (a) his ego and (b) his times. Both Bayesian and non-Bayesian (I'm not going to say Frequentist) approaches have advanced markedly, with quite a bit of convergence at the philosophical level, but proliferation at the model/methodological level. For example we have the bootstrap, empirical likelihood, MCMC, EM, AIC/BIC, diffuse priors/informative prior/non-informative priors, etc...so many ways to skin the same cat (but at least we all seem to agree on what it means to have skinned it!).
For (2), I actually think I partially answered this in my answer to (1). If you read modern Bayesians like Gelman, you'll see a strong push towards model checking and verification by comparing observed frequencies vs modeled frequencies or similar measures (e.g., he proposes a version of the P-value in his book for predictive model checking). The massive decrease in computing cost means you can also check a lot more measures of "misfit" than when Fisher and Pearson came up with their P-values and significance tests. Gelman and others seem to support the idea of holistic checks, not blind adherence to a single statistics. I agree with this, and I'm happy to see how objective verification has returned to Bayesian modeling.