Why are palladium and platinum carbonyls unstable at room temperature?
Tl, dr:  It's not just a matter of size.  A combination of coordination geometry, node distribution in the $d$-orbital wavefunctions, and atomic size all conspire to weaken $\pi$ overlap in palladium and platinum carbonyls, not only relative to the nickel compound but also relative to earlier transition metal carbonyls in their respective periods.
Ziegler et al. [1] studied the relative stabilitie of metal-carbonyl bonds for $\text{G6(CO)}_6$, $\text{G8(CO)}_5$ and $\text{G10(CO)}_4$ carbonyl complexes, where $\text{G6}$ in the first formula is a Group 6 metal and similarly for the second and third formulae.  The following general trends were observed:

Within a period, the G6 hexacarbonyl is most stable.

Within each group, the 4th period carbonyl is most stable, the 5th period carbonyl is least stable and the 6tg period carbonyl is slightly more stable than the 5th.


From these trends it is seen that palladium and platinum carbonyls are literally in a bad place.
The authors attribute the stability order primarily to $\pi$ backbonding.  We may expect such backbonding to be strongest for the G6 hexacarbonyls where the octahedral geometry about the metal atom causes the metal-atom donor orbitals to have pure $\pi$ symmetry.  The five- and four-coordinate geometries (trigonal bipyramidal and tetrahedral, respectively) are progressively less favorable with regard to this symmetry.  This accounts for effect (1).
Effect (2) is related to relative atomic sizes within each respective group, but also to a more subtle effect.  In the $3d$ subshell, which is used as the valence subshell in the 4th period, the lobes have no radial nodes, a condition that allows maximum overlap with the molecular orbitals coming from the carbon monoxide.  The $4d$ and $5d$ orbitals used by the heavier metals have radial nodes so poorer overlap.  With the 6th period metals, relativistic effects intervene, but this effect (which I cannot tell from the abstract, and the full paper is behind a paywall) is less potent than having more compact orbitals with no radial nodes at all in the $3d$ series.
Reference:
1.
Tom Ziegler, Vincenzo Tschinke, and Charles Ursenbach, &quot;Thermal stability and kinetic lability of the metal carbonyl bond. A theoretical study on M(CO)6 (M = chromium, molybdenum, tungsten), M(CO)5 (M = iron, ruthenium, osmium), and M(CO)4 (M = nickel, palladium, platinum)&quot;, J. Am. Chem. Soc. 1987, 109, 16, 4825–4837. Link