Interesting question! I'd never heard the distance explanation, just the tumah meit one, but never thought about the implications that you've pointed out. I wonder if there's an element of "can't make it worse"; the temple was already defiled, after all, and maybe their actions at least moved things in the right direction? But for a permanent solution rather than a patch, you need to do it right? Just guessing; I look forward to seeing sourced answers.

it also begs another question -- why didn't the Greeks loot the temple and take the solid gold menorah?

@rosends The version of the story I'm familiar with is that the Greeks did, in fact, loot the Temple. The Menorah used then was a temporary one made out of torches held aloft somehow (perhaps by kohanim, at least part of the time).

@Harel13 The story is that they used iron spears to make a makeshift menorah. Gold is only the best way required to make as specified in the Torah. However, if necessary other metals can be used if sufficient gold is not available.

do the rules of kedusha apply to a makeshift menorah?

I remember the same story: 7 days to travel and get new oil. No sources, so just comment and not an answer: If everyone (or majority) is Tamei then there are exemptions - which would apply to making oil as much as lighting the Menorah, but since there was at least one Tahor Cohen to light the Menorah, he could have made the oil - therefore that wasn't the problem. As far as cleaning the Temple, etc. that would be OK under the circumstances - again back to the everyone/majority issue.

As far as finding ashes of red heifer (not making a new one - that was extremely rare), presumably those were kept in a special place outside the Temple and retrieved as needed. As opposed to oil which was a commodity used daily and logically kept inside the Temple where needed (and therefore taken or bottles smashed, etc.) and also a common product used by everyone (just as olive oil is today) and therefore something the Greeks might take for their own use - they would have no use for ashes.

@sabbahillel Thanks. Yes, I know of the spear version too, but forgot about it. But I recall another version that used non-spear torches.

See https://judaism.stackexchange.com/q/3700

@Harel13 Your version is from Pesikta

@user8087 Highly appreciated. Shkoyech!