Thanks for your question. Project 2010 offers many options for organizations like yours trying to determine the best solution to meet user/technical/process driven needs.
Here are a few points. Let me know if you have any additional (specific questions).
1) We use Project Server 2010 to manage our service company and leverage the Project Management feature to the rich resource toolset.
2) The different roles you mentioned (managers, team leads, and supervisors) could all access Project/Resource/Task information via an intutitive webbased GUI.
3) You mention that Project licenses are tough to come by in your organization. Project Server 2010 allows web-editing of project plans so many less Project Professional licenses are needed.
4) With Project 2010 – users could create projects (in an Excel-like spreadsheet) on the web. These could be very quick and easy projects or they could leverage a template with an associated workflow.
5) I would recommend talking with an Microsoft partner (like MsProjectNow!) that specializes in Project and Project Server before ruling it out. Partner like us not only support the full server solution however have many customized solutions to fit every companies needs.
6) In addition you can view a webcast I did a while back titled Project 2010: Is An Upgrade Right for Me? – you can find the recording in the Knowledge Library under WebNLearns Recordings – just search my name.