Even easier, just take a limit of polynomial regression to a Gaussian process while optimizing the marginal likelihood over the prior temperature.
In all of these cases, the model with the least parameters is not the simplest in principle and does not have the best chance of not overfitting. The reality is significantly more nuanced.
Are you sure that doing this after seeing the data is valid and does not suffer from the equivalent of peeking-into-the-test-set problem ? There are ways to address the peeking problem but that requires additional machinery.
I don't dispute your broad claim but the first counterexample you quote seems problematic.
> and concentrate the likelihood around the zero loss set. Then reduce the variance on a Gaussian prior.
Those phrases could mean a lot of different things. What are you proposing?
> so that any measure of model quality will monotonically increase with model size and achieve a maximum at infinite model size.
any measure of model quality? You must have some bounds of any measure, since trivially that's false because "fewer parameters is better" is a measure of model quality, even if dumb.
It's hard to even engage when you're being so imprecise, and not even giving one specific example.
Concentrating a density around a zero set means that I raise it to the power of 1/gamma (appropriately normalizing) and then take gamma to zero. If the likelihood was Gaussian, this would be equivalent to taking the variance to zero (yielding a point mass). But in overparameterized settings, this concentrates on a submanifold describing the set of interpolating solutions. In least-squares linear regression, that is the solution space. Reducing the variance on a Gaussian prior is treated as an asymptotic expansion by Laplace's method. If you choose the variance to decrease (inversely proportional to the parameter size, for example), then the marginal likelihood will increase monotonically with model size.
By any measure of model size, I mean that you can pick your favourite among the common ones, such as information metrics (e.g. mutual information / KL), statistical metrics (e.g. marginal likelihood), test error. You should be able to show the same phenomenon happening for all of them, so it isn't a quirk of marginal likelihood. It is concentration of measure working in your favor to reduce the variance in the estimator.
E.g. over-fitting.
This is why the notion of overfitting is not nearly as cut and dry as a basic ML course would have you believe. Just because you fit data exactly does not mean that your estimator has high error on out of sample data. A trivial counterexample is a spiking model that spikes to fit to the data but otherwise follows the correct trend outside of the dataset. The bias variance tradeoff gets thrown out at enormous scale and overfitting is not a meaningful concept. What matters is regularization and robustness, not how well you fit the data.
The reason why bias variance tradeoff and considerations of model size are a good approximation for smaller models is due to concentration of measure in the data which effectively kills any regularization in your modelling procedure. Once you enter settings where concentration of measure begins to bite in parameter space, everything changes. This isn't really that mysterious; any textbook on Gaussian processes (e.g. Rasmussen and Williams) will tell you this.
Does model performance also concentrate into a 'typical case' where things work pretty well and a non-typical case where it's completely unpredictable as the number of parameters increase?