I don’t think that the examples the OP gives are the best ones - this sentence might be a better one.
“The engine doesn’t start” - in other European languages it would be “The engine starts not”. “I don’t think that…” would be “I think not that…”. In other words the use of the verb “to do” is superfluous since it doesn’t add anything (“it adds nothing”) to the meaning of the main verb, which in these examples is start, think or add.
The “language of the Saxons who occupied Britain” was actually Old English. As far as I know, like most other European languages Old English did not use this superfluous ‘do’ - it came into English several centuries later, and so was nothing to do with the Saxons.
I thought “do” as a modal was one of the few hangovers from Celtic languages in the UK. I remember this from university linguistic courses and a couple of threads on here that have refreshed that memory.
I can’t find anything online to substantiate this, however, since Google claims “do” is Old English, but Old English is not entirely made up of Germanic terms and grammar (if it were, it’d be called Old German), so saying it’s Old English isn’t very helpful. Do, in the way it’s used in English, to form questions, negatives and emphatics, is certainly not a feature of any other Germanic language I know of.
Anyway, “I did that” is emphatically (heh) different in meaning from “I did do that.” The latter is saying you did, even though there’s some expectation that you didn’t, and the “did” is usually stressed.
Like, in the Buffy zombie bit at the end of the credits, where a kid says “I made this!” that’s just a kid telling you joyfully that they made something. “I did make this!” means that’s a kid telling you they made something, and there was some doubt about whether or not they made it, but they did.
The “done told me” is a little different construction than the emphatic do. It’s what’s called the “completive done,” and is seen in a few American English dialects, including AAVE and Smoky Mountain. It does add emphasis to the completed nature of the verb. (Completive aspect.)
I suspect that when they say “do” is Old English they just mean the verb “to do” in its ordinary sense of doing things, rather than the sense we’re talking about here.
Other examples of what the OP is talking about are questions such as “Do you like the hedgehog?” rather than “Like you the hedgehog?” and “Where does she keep the orchidometer?” rather than “Where keeps she the orchidometer?”
Mammy Yokum to Abner, after his marriage to Daisy Mae: “Try an unnerstan, son. It don’t matter what Fosdick done done, or what Fosdick didn’t done done. Yo is hopelessly, irreversibly, incurably married.”
In the late 1970’s I was a sysadmin and had to write a brief weekly summary of what I had accomplished each week. I titled it “What I done done this week”. Everybody got a laugh out of that.
: “Do tell!”