syntactic-analysis
I'm not sure how to explain it, but whenever I come across a situation where two of the same words are used together I'm not sure whether, in terms of grammar, it is correct.
For example,
"I could get by in Malta by teaching the small portions of the Maltese population who can't speak English, English."
Is that correct to say, or has this happened because the sentence is worded poorly?
Would it make more sense if "who can't speak English" was a subordinate clause?
"Making sure we are on the same page." would be even better than the options you mentioned. Although I don't think your suggestions would be misunderstood, what I have suggested is a more common way of saying it. See below. 
Reference
I think the reason for your confusion is twofold - one is that your parsing of the sentence is incorrect, the second is a confusion over the way that "that" may be used.
To address the first issue - the meaning of the sentence is the number is staggering; the prepositional phrase "of people [who/that] don't give way to buses" is there to answer the question which number. Therefore, at the most basic level, the sentence can be broken up into:
[The number of people [who/that] don't give way to buses] (subject) is (linking verb) staggering (subject complement).
The subject contains a prepositional phrase that is also a restrictive relative clause people who/that don't give way to buses. Therefore, the phrase "don't give way to buses" modifies people, not number or number of people.
Now that we have worked this out, we can answer whether "who" or "that" can be used. As it is a restrictive relative clause that refers to people in an abstract way, you can use either that or who (although there are some who will argue that it is much better/only appropriate to use who, in practice both are used). Answers to this related question already give some explanation, so I will not expand further on it here.
For those interested, this Oxford Dictionaries blog post and this Grammar Girl article discuss the usage of who vs that.
Best Answer
This is a question of diction rather than correct grammar. It's awkward to repeat a word twice in a row unless you are reproducing the way someone actually speaks, in a line of fictional dialogue, for example. Simply reword it: for example, "I could get by in Malta by teaching English to the small portions of the Maltese population who can't speak it." There are a few instances in which repetition of a word twice in a row is familiar, for example: "Whatever is, is." (In that instance, the comma between the subject, "Whatever is," and the verb, "is," is allowed or encouraged.)