Hi Sean,

This is an interesting question and not a simple answer. The three verbs you mention are used for very different purposes.

The first one, used in the text, 'tenir', will be used for shops, restaurants, hotels, cafés and the best translation, in my opinion, is, 'to run something'.

Diriger is to lead ( to be at the head of something) so you might -

diriger une  équipe, une entreprise, un département, un service, une association, un organisme,  une agence, etc.

Gérer is to manage something, as in 'administrer'  often used with 'bien' or 'mal'.

so it can be used for -

gérer un budget, un problème, une émotion, un compte, etc.

It is interesting to point out that a manager in a large company with multiple departments will often be described as 'le responsable de ' and not 'gérant' which is used for small establishments where someone is paid to run the place for the owner.

Hope this helps!