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It means you can't easily (or statically) reason about the output given arbitrary (user) input. So yes, it makes it much more likely that security bugs are introduced.

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Also it at least implies a guaranteed resource leak / DOS capability, which is a problem though it may or may not be considered a security issue.

Well, that depends; you can slap a rlimit on a process and it will no longer DOS you, even if it's executing Turing complete code. It does mean that inside the Turing machine you can't really apply protections, but from the outside a Turing-complete language cannot magically escape its sandbox unless you give (or accidentally include) it a tool to do so.

Unless I'm completely off my mark, any resource limited system is not true universal Turing machine in the theoretical sense. So depending on your level of pedantry, TC can mean guaranteed DoS.

But...that's your whole computer too then.

Indeed.

For the purposes of this conversation, we generally speak of the TC-ness of a system if we could somehow expand it indefinitely.

I believe there was someone who set up a TC-complete roller coaster network in some Rollercoaster Tycoon sort of game. It had something like 25-slots of "RAM". Not very TC-ish. But if you could hypethetically expand out, hypothetically it would work.

So it's a common convention.

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