There absolutely are direct neutron experiments, but they are much lower energy and have a different focus, partly because neutrons being neutral means they’re very hard to accelerate.

There’s an ultra cold neutron source at Paul Scherrer that is used to measure if the neutron has an electric dipole moment. This is complementary to high energy experiments.

Neutrons are not that different from protons. The decay from neutrons to protons is pretty well understood, and there’s no reason to think that the nature of quark/gluon interactions in a neutron are significantly different from those in a proton. What kind of new physics are you imagining we’d get?

Of course more experimental data is a good thing, but in this case it doesn’t seem obvious that it would lead to anything really new.