When conveying a strong desire or demand, there are several synonyms for the phrase "insisted upon". One could use "asserted" to imply a strong and confident expression of a need or desire. "Emphasized" can be used to indicate a particular point that the speaker considers to be essential. "Pressed" implies a sense of urgency or a need for immediate action. "Demanded" is a forceful term that suggests a strong necessity to fulfill a request. "Urged" can be used to indicate a persuasion or appeal to take a certain course of action. The choice of word will depend on the specific context and desired level of assertiveness.