We can get probability $\ge 70\%$ with six 10-sided dice:

 - 10 sides $=6$
 - 3 sides $=11$, 7 sides $=5$
 - 4 sides $=10$, 6 sides $=4$
 - 5 sides $=9$, 5 sides $=3$
 - 6 sides $=8$, 4 sides $=2$
 - 7 sides $=7$, 3 sides $=1$