In fact, that's how the interstates work too - the state transportation agencies apply for highway funding from the federal government and then manage the projects.
Although I'm not really sold on even the approach you describe - it seems like it's just a way to funnel taxes from local people -> federal -> back to local.
Skip the middleman?
California tax payers support West Virginia infrastructure.
Inevitably some states are wealthier than others, the federal government acts as a balancer of this. This can improve outcomes in poorer states (education, building up new industries, local economies etc).
Ideally, it also provides a counter balance for changing economies, where there are inevitably winners and losers. Many industries are geographic centered, and having the ability to adjust for acute downsides in one area benefits the whole country overall in the long-term.
If it's not worth funding locally, it's not worth funding federally.