Most agencies treat the intake form as a contact tool. Name, email, company, what are you interested in, thanks for reaching out. The shops that are running their week better treat it as the proposal brief, because that is what it secretly is.
What the intake form is actually doing
By the time a lead fills in an intake form, they have decided three things you cannot get from a discovery call. They have decided what to call their own problem, what shape they want the engagement to be, and what they are willing to put in writing as a first cut.
A short, well-built form captures that snapshot. A long, generic form throws it away.
The agencies that win their week use the form to start the brief, not to gather a phone number. The proposal that gets written next week is half-drafted by the form that came in this week.
What goes on the form
A short list. Long forms get abandoned, and the fields you wanted most are blank.
What problem you are trying to solve. One open question, three lines of space. Buyers will write more than you expect when the question is framed as "tell us the situation" instead of "what services are you interested in."