Rather than creating a "free alternative for MathSciNet", it seems more useful to do what MathSciNet does not, e.g. make a site that maintains lists of errors in published papers.
I personally think MathSciNet is a great product, very much worthy of support, and I do not get the argument "since MathSciNet is not freely available, one should not support it". Just stop using MathSciNet for a few months and see how it feels. Finally, posting your reviews on your own homepage will make them publicly available; I see no problem here as well.