I wish I could give you a definitive "yes" and have you start working right away, but with most questions like this, it depends on a bunch of factors. I'd first like to say that any advice I give here is limited because I've never seen your current contacts solution and what it gives your team already.
If possible, I would like some help understanding the terms you are using. Can you do me a favor and describe exactly what each of these things mean to you and your system:
Even before getting full descriptions of those, it seems like what you need is a more relational system with multiple tables of data that relate to one another. This is obvious when you say things like:
editing one record rather than several
a person may belong on several committees
What those relationships are, I don't know right now. Either way though, I can tell you that similar to your file, none of the Starter Solutions have a relational contact solution like what you are asking for. The Contacts has only one main contact table, which is probably similar to your solution. You mention invoicing too, which might mean that you want to think about using the Invoice Starter Solution, but the "contact" piece of that solution only has Companies, so you would potentially need to build out the contacts/departments/committees part.
So whether you use our starter solution or your old solution, you will need to build out a more relational structure. This means spending some time figuring out the right data model, building that in Manage Database, then creating layouts, scripts, etc... to deal with this new structure.
As for the online directory, I know both Contacts and Invoices have FileMaker WebDirect optimized layouts, so that might be a nice starting point.
Since you'll probably have to do some building, it depends on what you feel more comfortable doing: building in something that you know in and out, but might have a lot rust and detritus, or building in something new, but that has a learning curve because you don't know it in and out.