Do you feel the older you get, the more proud you are of your service?

I don't think more pride comes with age. Pride is there or it's not there. What makes it look like there is more pride with time is that there are less people around having pride in what they do/have done.

Anyone Who Served Honorably Should Be Proud. The S-1 Clerks, The Cooks, The Medics And All The Other Support Jobs Kept The Front Line Troops In The Battle.
They Did A Job That Most Americans Didn’t Do.
Even During So Called Peace Time Any Of Us Could Have Given Our Lives For This Country.
That Alone Is Something To Be Proud Of.

Proud of my service? Most certainly.
What I don't have is an attachment to any one unit. I served in 4 divisions, 2 Corps, 3 Support Commands, and 2 hospitals. I started in aviation, after 'Nam, became a mental health counselor, and worked in clinics.
I don't know about now, but back in the day, the mental health / drug -alcohol rehab folks were not really welcome company at social functions. Folks were afraid of us.
Just about everybody I saw in my work was in trouble, and a half-step from a discharge.