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I've noticed that some groups of metals with similar properties tend to belong to the same cubic structure group, such as $\text{FCC, BCC, or HCP}$. What is a good explanation as to why these metals have similar cubic structures, and how can one determine a metal's expected cubic structure just from knowing its identity (if there is a way)?

For example, copper, silver, and gold all have face centered cubic structures.

This trend seems to break down at some point, I think, because $\frac{2}{3}^{rd}$ of the platinum group have $\text{FCC}$ structures and $\ce{Ru}$ and $\ce{Os}$ have simple cubic structures.

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There is no systematic way (besides brute force - that is a system, right?) to predict the crystal structure of the metals. Even sticking just to the transition metals, as you have noted, there are quite a variety. The primary ones are as you have noted $fcc$, $bcc$, and $hcp$ (ignoring the rhombohedral Hg).

Several issues arise here. First, there is little difference between $fcc$ and $hcp$ - they are both close packed structures with the only the stacking order changing (ABCABC for $fcc$ and ABABAB for $hcp$). An estimate of the free energy difference between these two crystal structures can be obtained 'SGTE Data for Pure Elements', A.T. Dinsdale, CALPHAD 15(4) 317-425 (1991), the compilation of elemental thermodynamic data. Here one finds that for Ag ($fcc$), the $hcp$ phase is only 300J/mol higher free energy (at T=0K). Similarly, for Co ($hcp$), the $fcc$ phase is only 427 J/mol higher. So, the free energy differences aren't necessarily that much, making an $a\ priori$ differtiation impossible. (I'll note here that Ru and Os are $hcp$ crystals, not simple cubic as you stated).

The second factor that is difficult to determine is the impact of magnetic free energy on the equilibrium crystal structure (as included in the above reference). Iron is the most obvious candidate to consider this, and it does not disappoint. As observed, the equilibrium crystal structure at room temperature is $bcc$. Heating leads to a change to $fcc$ at $910C$, then back to $bcc$ at $1400C$. If the magnetic contributions to the free energy are ignored, the observed crystal structures would instead be $hcp$ at low temperatures, a transition to $fcc$ at around $200C$, and a brief appearance of $bcc$ right before melting. I see no way that one could predict these particular changes in phases with temperature.

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