"insisted on speaking" is a grammatically correct and commonly used phrase in written English.
It is typically used to describe a situation where someone is determined or persistent about wanting to speak or be heard. Example: The child insisted on speaking to her teacher about the bullying she was experiencing.
It's a similar story with the MP to whom I picked up the phone years ago – back when I was the only woman on a team of five political reporters – who loftily insisted on speaking to "one of the men".
When a shareholder at the Basel meeting insisted on speaking in Italian, one of Switzerland's four official languages, Mr. Ospel cut him off.
"Her posture was to sort of challenge — not so much the president — but his advisers, whom I was one," said the official, who insisted on speaking anonymously.
"There is a political movement because of the financial debacle," said one veteran European banker who insisted on speaking anonymously because he has retired.
Betty LaFrance, who was not one of Gordon's students but always insisted on speaking to him, brought a radio from her cell and played elevator music as she crafted.
But that was because of a happenstance confluence of religion, the birth of a nation and the obsessive devotion of one man, Eliezer Ben-Yehuda, who settled in Palestine and insisted on speaking only Hebrew to all Jews.
If, on the other hand, one insists on speaking strictly, then (Lewis admits) one is forced to acknowledge that there are desiderata of moral values (such as the authoritative practical oomph that Mackie goes to such efforts to articulate) that these dispositions do not satisfy.
Today, one can glean only non-nourishing tidbits from designers, who insist on speaking off the record to protect their business with her.
