How do I start level 70 Summoner quest?

[MistakeNot]

I wouldn't say "not at all," LOL - by the time my level 60 job quest is available, I'll probably be level 73.

No need to outlevel content by that much. If you use multiple jobs to share the xp, you can stay close to on-level all the time. But it is your choice how to play the game.