Turned outHmm, I've found that if I needed to add ".value" to the end of the variable assignment, the script works. However, if I move the script to a Static Resource, it no longer works.
i.e.:
var override_plat = document.getElementById('{!$Component.page1.block1.form1.overr_plat}').value; //works in <script> tags on VF page, but not in Static Resource