Restricting yourself to a small bullet is like shooting a solid or a hardcast that does not expand..A .223 expands to a caliber size larger than all pistol calibers and 99% of rifle calibers using a non expanding bullet.
I have Elk hunted since I was 2 years old and started shooting at 10 with my first Elk at 11(memory)..Elk hunting does not mean what it used to to me..I enjoy getting out/teaching my kids and seeing game but lately it is just me and the wife and I don't get down and dirty after them like I used to,not because I can't, but because I really don't want to anymore.
Yeah,I am going to hunt with the .223 next year as a stunt maybe or a "challenge" mostly.We had a 6X6 walk right through camp last year in my sons unit(not mine) and watched it broadside for maybe 90 seconds,time enough I could have grabbed a rifle and blasted it as it stood there less than 100 yards away broadside.
I know where there hanging out but I couldn't go in after them out of my legal unit.
In answer to your question,the smallest caliber I have used on Elk is the .270 and between myself/my grandpa and my dad,I have seen an easy 100 fall to the mighty .270 Win..I started out as a packers son never missing a season and never being skunked.Times have changed since the old days...But life goes on and I love a challenge,a real challenge..Keeps my blood moving.