The core (in my opinion) is the foundation of all movements. Just as you need solid foundations for a house, or a stable base. The core can certainly be influenced by foot bio mechanics, knee position, hip position upper body position etc however no matter what sort of injury you have, foot or otherwise I believe you should start at the core first.
The deep core includes the transverse abdominus/pelvic floor/multifidus. It is not necessarily a strength issue a lot of the time just an activation issue. You simply need to learn how to wake these muscles up and integrate their use into every day life.
The Gluteals are often spoken of being the most important muscles for stability- and yes they are very important for posture and efficient movement. However, I believe their effectiveness is limited by core function and this is the reason I highly advise pilates taught by physios and I have a great network referral locally for these types of classes.