I have cross-read the Wilkinson articles. He describes how to design low pass FIR filters that have a monotonous (i.e. not rippled) passband and which do not distort polynomial signals up to an given degree (of length at most the filter length). In the first paper this is done for symmetrical (i.e. linear phase) filters. In the second paper he does it for asymmetrical filters, keeping the phase in the passband next to linear but does not care in the stop band. Moreover he describes how to shape pass- and stop band if you do not want it "flat".

Parks-McClellan in contrast has (by design) passband (and stop band) ripple. You can bound the passband ripple by design to any number you want.

Wilkinson mentions that in his filters, when implemented in real, due to arithmetic precision, a very small passband ripple may be introduced.

In both, Parks-McClellan and Wilkinson, the filter coefficients are given implizit, as solutions of a set of equations. In general they are computed by numerical approximation. If you make the requirements special enough you may be able to solve the system explicitly, i.e. get the coefficients in closed form.

In light of this, for Mike Moffatt closed form filters, "they" must have made enough restrictions to be able to solve the coefficient equations explicitly (even if they use an other approach as Parks-McClellan or Wilkinson). Restrictions must not be a bad thing if they are adequate for the problem.

However I wonder: Wilkinson's articles are 20 years old. He had an implementation of his design algorithm at that time. There must be available filter design algorithms that use Wilkinson as an option or are that general that the "Wilkinson-type" filters are included as a subcase.