Please clarify what you mean by “give all your medical info”.
Although this article concerns the fedgov exchange website, my state’s site required only that I give the ages of myself and family members and monthly gross income to get the premium and subsidy. There were no questions of any kind about medical history there or in the full application process (other than “Are you a smoker?”, sort of).
Once you gave them you name, date of birth and ssn they had access to your IRS and any on-line medical information. The reason that these systems are so slow is that they link to the IRS, Medical, Federal criminal, and other databases.