Did the "We shall fight on the beaches" speech mainly use words from Old English? If so, why?

To take the best-known passage alone, rather than the whole speech, and skip over duplicates, we can quickly show that this is the case:

we Old English we

shall Old English sceal

fight Old English feohtan

on Old English on a variant of Old English an

the Old English þe, from earlier Old English se

beaches Old English bæce/bece

landing Modern English noun from land, from Old English land/lond

grounds Old English grund

in Old English in

fields Old English feld

and Old English and/ond (in which it at first only had the "then","next" meaning rather than the "and also" meaning).

streets Old English stret in some dialects, stræt in others.

hills Old English hyll

never Old English næfre

surrender Old French surrendre

Now, the next question is to what extent does it tell us something about English, and to what extent something of Churchill's mastery of it?

The speech is in general not one that attempts to demonstrate Churchill's intellectual mastery of a subject (to which we turn more often to words of Latin and Greek origin), or which requires technical precision (in which we turn to the terms with most precision in a field, which tend to be Latin or Greek again), or his florid emotions (to which we turn more often to the Romance languages, which in turn owe a lot to Greek). No, he wanted to underline a strong, almost crude drive, simply expressed.

The speech called for simple words, and the simplest words in English - those that are known to everyone from a young age - are mostly Old English in origin. Old English is the core of English after all (that's why it's called Old English). Even a speech that made heavy use of Latinate words, would have much that came from Anglo-Saxon (all the articles and pronouns for a start).

Even if he was aware he was using Old English words heavily (and as the historian whose works included the four-volume A History of the English-Speaking Peoples, it's quite likely that he would have been conscious of it), his primary concern was more likely this simplicity. He was not going to make an intellectual argument in favour of continuing the war, he was going to drill in a series of similar emotive assertions that would encourage the hearts of his fellow politicians, his fighting men, those otherwise involved in the war-effort, and the general populace, alike.

And most of us would turn to mostly Anglo-Saxon words in such a case too.

He may have been considering the emphasis upon Englishness suggested by this. He would probably not have wanted to do this too blatantly. After all, the allies where not solely English (in the context of the speech, they were mainly French), and even those in British uniform included many from elsewhere in the UK and its colonies, along with some volunteers from elsewhere. I think, too much is made of this possibility.

So too, do I think too much is made of the word surrender being of Old French origin. In recent years much has been made of this, particularly in response to some in the US attempting to stereotype the French as militarily weak after the French government opposed the invasion of Iraq. Aside from ignoring just about everything about French and US military history, it's doubly irrelevant here.

For a start, the speech was made during the Battle of France. The BEF were fighting along-side an International force that was mainly French, to keep France free from the German invasion. This is not a time where Churchill would have insulted the French!

Second, we can consider the effect of this word by changing "We shall never surrender" to "We shall never give up", which would make the passage entirely Old English in origin, but the words "give up" differ from "surrender" in both nuance and euphony. In addition to being almost all Old English, the words are almost all short, being mostly monosyllabic with just two two-syllable words (one of which is part of a three-syllable noun-phase). The three-syllable surrender allows a slightly longer consideration of that assertion — even though it's a negative one — of what they would not do, coming as it does after his previous positive assertions of what they would do.

In all, I think the heavy use of Old English says more about English than it does about Churchill's use of it. This is not to deny thatChurchill was a master of the language, and his speech was a great example of that mastery. Many other eloquent speakers and writers of English would similarly have used Old English heavily if they wanted to evoke a similar response in their audience.

It may be true of those few famous sentences, but a cursory examination suggests that it isn’t true of the whole speech. I don’t think it’s a case of his deliberately choosing words of Old English origin. It’s just that such words happen to be in normal use for the things and actions he is describing. How else was he to say We shall fight on the beaches? No one would have expected him, or anyone else, to say We shall engage in pugnacious activity on the littoral.

What Robusto said BUT as I am somewhat of a Churchill mind reader (yellow-belt, 2nd class, wades boldly through deep puddles), I can add confusing opinion on the second part of your question.

Churchill was a master of stirring rhetoric, flowery turns of phrase, clever descriptions and memorable phraseology. He was also very much aware of Anglo-Saxon history, the source of words and the historical relationships of all the protagonists in the coming affray. Had he sought to use solely or mainly Anglo-Saxon words he would have been about as capable as any orator of the day of doing so. However, I do not think it likely that what he did (if he did indeed do it) would have been intentional. While it would be conceivable for him to have wanted to "send a message" by restricting himself to olde language with certain historical connotations, it is much the rather likely that he would have been aiming at maximum impact, maximum resonance with national pride and values and maximum appeal to "all that was English" in the hearts and minds of his hearers. If indeed Anglo-Saxon does form the major part of the core of the English language, then a "back to basics, national-fundamentals and historical-loyalties, blood-sweat-&-tears message, such as Churchill would have been seeking to convey, would reasonably be expected to be largely couched in core terms, and in turns of speech which resonated with an Englishman's expectations and sense of pride in his country.

If Anglo-Saxon does not in fact form the majority of core English then none of the above will be true :-).

AND it may not be true anyway.
Of course.

But, knowing Churchill, and knowing English, odds are that this is as good an explanation as any you'll get.

Footnote / aside: The famous BBC version of Churchill's speech, which one can still hear recordings of, was NOT delivered by him. He 'got a bit busy wit related matters' immediately after giving his House of Commons version and come BBC broadcast time he could not afford the time. An actor who was experienced in Churchill's style gave the speech. Churchill was subsequently both amused and impressed at how the had managed to replicate a number of Churchill's mannerisms in his speech.

If you won't accept the authority of Robert Lacey, then I'm afraid nothing anyone here could say will persuade you. Short of an analysis of each of the ~53 unique words in the text—more work than is warranted for a question on this site, surely—no one will be able to tell you authoritatively if the claim referenced in your first question is true. I personally would not be inclined to doubt it. If you want to get out the OED and research them all, though, you should have your answer.

What I can tell you is that the statement about Churchill in Lacey's (excellent) book is not documented. The closest documented declaration immediately preceding the quote asserts only that "the hundred most frequently used words [in the English language] are all of Anglo-Saxon origin." This is derived from The Story of English, by Robert McCrum, Robert MacNeil, and William Cran (1992).

As to your second question, you seem to be asking us to interpret Churchill's intent vis-à-vis using Anglo-Saxon words in that speech, which is a task beyond us unless we were historians able to find such an assertion in Churchill's own writings. As such, the answer to such a question does not really belong on this site. Perhaps Skeptics.SE would be more in order here.