Imagine a lump of a metallic element in close contact with an equal amount of liquid mercury. Amalgamation might proceed to some degree if the metallic element has an even higher surface tension than mercury. A surface layer of adsorbed mercury will be the first step in amalgamation.
This first step, wetting, but not involving extensive solution of, or into, the metal, is well illustrated by this image of a mercury switch (Ref 1). The electrodes are well wetted, but must remain integral for many years thru electric currents, temperature changes and sparking.
Mercury can penetrate some metals, like aluminum, probably faster thru grain boundaries, thereby causing the metal to disintegrate. There is an added complexity that aluminum has an impermeable oxide layer that must be breached first.
Atoms may disperse thru mercury until a solubility limit, or more accurately, a viscosity increase is reached that effectively immobilizes the mercury. The term amalgam commonly means a fluid or semisolid material like a dental amalgam, which, although it finally becomes hard enough to bite on, is usually encountered in a semisolid state (see the image below, Ref 2). The time required for complete dispersal of the mercury in the liquid into the solid particles of the silver-tin alloy determines the alloy’s usefulness.
The phase diagrams of other mercury alloys shows the effect of melting point of the non-mercury element. The thallium (mp = 304 C) alloy is liquid at room temperature to about 45% thallium (Ref 3).
Uranium, melting at 1132 C), can stiffen a 1% alloy at room temperature, and 3% uranium will solidify the alloy as the mercury boils away at 357 C! (Ref 4)
Another way amalgams can form is by addition of the other metallic element, atom by atom, one at a time. Such is the usual method of preparing sodium amalgam electrolytically. A pool of mercury in contact with a solution of a sodium salt is made negative enough to attract sodium ions, where they will discharge and enter the mercury mass before reacting with water. This can be continued until the mercury is stiffened, which amounts to reaching a solubility limit, and at that point, any more sodium formed will react with water.
Aluminum and sodium have relatively low melting points, suggesting that they have low cohesive strength. “Low melting point” also includes silver and gold, with melting points around 1000 C. These metals can be expected to alloy, or amalgamate, easily with mercury.
But high melting point (for Pt, 1768 C) is not a guarantee of complete non-amalgamatability. The OP mentioned a way of making what appears to be a soft solid at 12% Pt from Pt sponge, which can be stiffened with more Pt, but this cannot be done with Pt foil. Pt foil can be obtained from Aldrich as thin as 0.025 mm; Pt sponge is composed of particles less than 10 microns thick - i.e., with 2500 times the surface area (Wikipedia). A surface reaction product Pt(Hg)4 has been found, but no penetration into the Pt (Ref 5) until the temperature is raised to 250 C (Ref 6), when the solubility reaches 15.5%, but forms another solid, not liquid, phase.
X-ray analysis showed that V, Nb, Ta, Ti, Zr, Cr, Mo, W, Fe, Ni, Co, Re, Ru, Rh, Os, Ir, Al, Th, and U dissolve only negligible amounts of mercury (Ref 6). So melting point can be a good indicator of the ability of a metal to alloy with mercury. Higher melting point suggests greater difficulty in atomizing the metal. There could be a correlation with surface tension of the metal. And one more quantitative measurement should be the heat of vaporization (not fusion, since the heats of fusion are quite a bit smaller). The heats of vaporization of tungsten and tantalum are higher than all the other elements I have checked by a factor of about two! (WebElements)
Temperature of preparation is important because some mercury-metal alloys may form only at higher temperatures. Whether the amalgam is liquid or solid will depend on the concentration of the metal in the mercury and the observation temperature. And time may affect properties like fluidity and stability (e.g., note the instability of ammonium amalgam, although it is easily prepared).
Ref 1. https://en.wikipedia.org/wiki/Mercury_(element)
Ref 2. https://pocketdentistry.com/14-silver-amalgam/
Ref 3. https://www.researchgate.net/figure/Phase-diagram-of-binary-system-Hg-Tl_fig3_47393888
Ref 4. https://www.researchgate.net/publication/238894897_The_HgU_Mercury-Uranium_System
Ref 5. S. K. Lahiri and D. Gupta, Journal of Applied Physics 51, 5555 (1980); https://doi.org/10.1063/1.327440
Ref 6. G. Jangg & E. Lugscheider, Monatshefte für Chemie / Chemical Monthly 104, 1269–1275 (1973)