How will your employer treat your pay and benefits status while you're on leave?
Disability income coverage and leave policies work in tandem to solve very different problems. Disability income coverage covers your income, leave policies guarantee your status as an employee.
Typically, STD coverage requires an actual loss of income and will offset it's stated benefit for any income you're receiving. In general you can't begin a STD claim after the 7 day waiting period and also draw income from vacation or sick time. Also, typically STD will cover some percentage of your covered pay (sometimes including commission/bonus income) up to some weekly maximum.
FLMA requires employers to allow certain amounts of time for certain types of leave. FMLA is not necessarily an income replacement tool like STD coverage. Contrary to your post it's my understanding that if sick and vacation time accrue in to a single PTO bucket your employer is prohibited from requiring employees to exhaust accrued time prior to beginning FMLA leave.
In general, you're not missing anything because the point of FMLA is to guarantee your job and status as an employee from a benefits perspective.
Benefits language from the Department of Labor Website
A covered employer is required to maintain group health insurance coverage, including family coverage, for an employee on FMLA leave on the same terms as if the employee continued to work.