English learners are taught not to write "the" when writing dates but I see that many native speakers do. Or am I wrong? The sentence below was written by a native speaker. "I paid off my PayPal Credit balance on the 10th September."
As a native speaker, I was taught the same as you - that the words "the" and "of" are pronounced, but not spelt out. So I would write:
On 10th September
(Traditionally, the "th" is written in superscript - 10th. With the decline of handwritten correspondence and the rise of print, this practice nearly disappeared for a while, but has returned now that many word processors will do it automatically for you. It is nevertheless optional.)
It is equally acceptable to write:
On 10 September
Both are pronounced "on the tenth of September".
This is standard practice (to omit the "the" in writing), but that doesn't mean that all native speakers will follow it. In formal usage, "the" can be reinserted, but "of" must be as well:
The more complicated the style of date, the more formal it is... The and of are optional, but if you do use them you must add both the and of; it would be incorrect to say only 13th of April or the 13th April. (English Lessons Brighton))
If I were writing out the date in full (as words rather than with figures), I would write:
On the tenth of September
but we rarely do that. It might be appropriate in a legal document. I also agree with Michael Harvey that it would be better to use the full version in direct speech:
"I'm seeing him on the third of October," she said.
The full version would also be common when referring to certain holidays or festivities (in which case the ordinal number would probably be capitalised too):
On the Fifth of November
On the Fourth of July
I would also include the "the" if I were omitting the month:
I was there on the 10th.
(This is another place where "tenth" would work well.)
Much of the above is primarily applicable to British English. In American English, except in very formal contexts or when referring to the Fourth of July, they write dates the other way round - as "September 10th" or "September 10" (although nowadays including the ordinal suffix is apparently rare in the US). British people sometimes do this as well, although some Brits regard it as an American practice to be avoided. Here the word "the" is not written, either, but is still pronounced - at least in Britain; I have observed that Americans often say "September tenth" without pronouncing "the".