MI6 doesn't stand anywhere in the "hierarchy of the government", because it is not a member of the government, though it is a section of Ministry of Defense. The MOD is a wing of the executive, under the theory of the divisibility of political power, but it does not occupy some sort of protected position whereby its members enjoy absurd and unnecessary luxuries. "Public sector" does mean the same thing for all of the institutions it describes: it describes a public service paid for by the Treasury and ergo by taxpayers' money. They have to budget thriftily and submit detailed accounts to Treasury auditors. While you might be correct to speculate that the intelligence agencies receive extra funding from other wings of the executive, I fail to conceive of a scenario where this would be spent on a massive telly.
Regardless, it is just a stylistic flight of fancy that I happen to dislike. You're perfectly free to like it, obviously.