What is the usual pattern of use of a velodrome (cycling track)?
Velodromes are not a suitable cycling venue for members of the general public. They require specialized equipment (e.g., higher bottom brackets to avoid pedal strikes), training (e.g., no moving up/down track without ensuring you have room, maintaining enough speed in corners), and discipline.

Expecting untrained riders and/or unsupervised novices to ride on the track is practically begging for crashes, injuries, and lawsuits.

As an example of restrictions on use of public velodromes, the Dick Lane Velodrome in East Point, Georgia was built as a public venue. Control over operations was granted to a 501(c)3 nonprofit, the East Point Velodrome Association. It is operated as a sporting venue. Members of the public are allowed to ride on the velodrome during unscheduled hours provided they have signed a release waiver, are accompanied by at least one other person, and have completed a weekend-long training course. Track bicycles are required, with the single exception of pursuit-style events, where riders may (but for all intents and purposes never do) ride a standard road bicycle.

Cheaper and more appropriate for your purposes would be to petition the city to build a circuit. Essentially, just a mile-or-so long, limited-access, paved bicycle path that loops back onto itself. No banking necessary, and bike paths are cheap and fast to build. You're not constrained to a flat oval, and turns need not be sharp enough to require banking for fast riders.