Disclaimer: This isn't really the answer Jan is looking for.
Holleman, Wiberg "Lehrbuch der Anorganischen Chemie", de Gruyter, Vol. 101, notes that:
Selenium
Selenium forms a weaker, but more stable acid based on oxidation state $ +4$. No explicit mention of the structure, which suggests to me that $\ce{SeO(OH)2}$ is the structure.
Tellurium
Tellurous acid's structure is described as unknown.
Arsenic
For arsenous acid (the $+3$ oxidation state based acid) the tautomeric equilibrium is entirely on the side of $\ce{As(OH)3}$ (as opposed to $\ce{HAsO(OH)2}$). The reason given is that $\ce{As}$ does not form double bonds with $\ce{O}$, which is outdated: Current thinking is that only the first row forms double bonds, i.e. no double bonds in $\ce{H2SO4}$ or $\ce{HPO(OH)2}$. Organic esters exist that can be described as $\ce{RAsO(OH)2}$.
Antimony
The $+3$ oxidation state based acid is described as forming $\ce{H+ + [As(OH)4]-}$ in water. No mention of tautomerism.
The book gives sources only sparingly and on a per-section basis. I may add them later.
My thoughts on $\ce{As, Sb}$ are that in the MO picture, $\ce{H}$ has insufficient overlap to form a bond that would compete with the alternatives.
I have performed DFT calculations (PW6B95-D3/def2-QZVP//PBE-D3/def2-TZVP) on gas-phase $\ce{Pn(OH)3}$ and $\ce{HPnO(OH)2}$. Of course, these are gas-phase energies on single conformers, not free enthalpies. For $\ce{Pn} = \ce{N}$, a 10 kcal/mol favorable energy difference towards $\ce{HPO(OH)2}$ was found. For $\ce{As}$, about 35 kcal/mol towards $\ce{As(OH)3}$ was found. The Mayer bond order analysis was not conclusive.