I have had this same problem in the past but I can't recall how I solved it. When I display your file, I get dots in all the various cities just as you would like. This would indicate to me a slightly different Tableau setup rather than a problem with configuring the visualization.
The only difference I see between yours and mine is that I have lats and lons as DIMENSIONS. Just drag the Latitude (generated) and Longitude (generated) measures up to the Dimensions area. That should make tableau create an axis for each which is what you want. Good luck.
Awesome. Thanks for the help.
Better ways to do this are to put a primary key on the level of detail shelf if you have one (you appear to already have Site # there) or turn off Aggregate Measures (in the Analysis menu). Either way assures that each record is treated as a separate point.
Looking at your file, I noticed you have State on the Level of Detail shelf, but don't have its geographic role set to State (Abbreviation). Your map shows a point for Newton in Kansas instead of Massachusetts because Tableau doesn't know to use the state field to pick the right Newton.