SE isn't going to also boost optional side content as part of your character boost.

Your chocobo is always the same level as your currently active job. You're not talking about its level. You're talking about its rank.

If you're already into level 70+ content, it should be ranking up very fast as long as you're killing things. The amount of chocobo XP from that content is considerable compared to what's needed for the lower ranks. Take an hour or so to do the multi-creature FATEs in Lakeland (don't bother with NM FATEs as your chocobo would only be getting XP from one kill).

There's nothing you need to do for MSQ that requires a chocobo companion. It's all designed to be solo outside of dungeons/trials. If you're having problems killing things, learn your job and make sure your gear is up to date.