Has any branch of the military made legitimate progress towards integration?

The Coast Guard tried some specially-named program back in the Nineties that was an utter fiasco. There still exists a broad chasm between the active and Reserve sides. Don't even get me started on the overwhelming majority of agents who comprise the ranks of the Coast Guard Investigative Service and how they remain bastardized!!!

Integration of who specifically?

The Army is formalizing this now. We have had ad-hoc partnerships between the ARNG and the Regular Army for a long time. "Total Army Force Policy" is the actual name for it. The Army has doubled down on this now that GEN Milley is the CSA though. We are now "associating" units from the ARNG to the Regular Army and vice-versa. This is a different kind of relationship where the unit associated actually has a significant ADCON relationship with the gaining unit. There are significant responsibilities which shift between the ARNG and the Regular Army. The Army will likely have the most formal and integrated service once this takes full effect.