Here's what I see: you have identified a problem, and the problem is so big that you are itching to build a solution.
This seems to be independent from your question. Everyone on HN can come up with a reasonable speculative answer to your question (and mine would be a "yes"), but focusing on the issues makes clear if it's even worth competing in the first place.
With the information you provided, you seem to have a solid target market identified. How many teams are there that, like yours, that could benefit from a Basecamp alternative? And which of your team's needs aren't met with Basecamp, and is it something that Basecamp could replicate quickly? Dig a little bit deeper and discover the specific needs - that should reveal more useful information than any of us could provide :)