Since 11C SSGs aren’t currently being selected for DS duty, what are some other broadening assignment options?

It won't stop your career progression. You're competing against other SSGs for SFC, so everyone else is in the same boat as you.
Other options: Ranger School, RI, AIT or ALC instructor, SAMC, Black Hat, JM instructor AC/RC.
A lot of people in the Army are disqualified from DS and Recruiting due to stupid mistakes they made while younger. There are plenty of other opportunities to pursue.

It was just the opposite in the late 90's and early 2000's, 11C SSG's were over strength and we were detailed to recruiting duty by the truckload. You did your 3 year tour and returned to being an 11C, then your SFC promotion packet went to the 79R stack at the selection board because there were far too many eligible 11C SSG to promote in that MOS. I was an 11C, made the SFC list twice as a 79R. I declined it the first time, I didn't want to give up being an 11 Chuck and go back to that hell in USAREC. The next year I was on the list again, and after being counseled by the BDE CSM and understanding that the board was rigged for the "needs of the Army", I took it. That was 2001, literally 3 days before 9/11. When I went back to recruiting school, 3/4 of my class were former recruiters and newly promoted SFC's. Many of them 11C's, including a buddy of mine who was also one of my IMLC instructors just 6 months earlier. My advice to you would be avoid volunteering for recruiting. If you happen to be successful and survive that assignment, there is good chance that "needs of the Army" may result in your career being detoured in a way you never intended. The Army Recruiting Command always struggles to fill 79R Cadre slots because there are never enough volunteers to re-class. MOS's like 11C are always a target for conditional 79R E7 promotions because of the limited SFC and MSG slots in that MOS. Chances are though, "needs of the Army" will catch up to you eventually and you will get detailed to USAREC. If you want to avoid all that, don't give up on your DS packet. Good Luck.

Interesting! Have you tried a CTC assigmament as an OC? NTC/JRTC. Not only broadening; and if played right family quality time increases.