Of course it is interesting, and the idea of factoring out symmetries goes back to Newton.
As for your point (1), yes it leads to more complicated geometry, and often some form of singular reduction is required (e.g. have a look at the blue book of Bates & Cushman for the case of integrable Hamiltonian systems with finitely many degrees of freedom), but from a dynamical point of view it makes much more sense.
For instance suppose you would like to study numerically a classical mechanical system (integrable or non-integrable all the same): working in reduced coordinates allows to easily distinguish between different Periodic Orbits and Relative (i.e. to count them only once), while in non-reduced dynamics the Periodic Orbits come in continuous families.
As for your comment, which would not be so obvious simply looking at plots of the orbits in topology of the original coordinates (reduced space is certainly a "new physical insight". Have a look at this recent paper by Yanguas, Palacian, Meyer & Dumas where one could very well confuse an RPO for an actual PO since they all closestudy periodic orbits of a non-integrable system as you ask, while RPOs don't close in and where they discuss the symmetry-reduced dynamics)issue of reduction and provide further references.
(Edited to correct an error; more links added in reply to comment.)