Currently I bring to class a variety of different style tug toys and her dinner of kibble. During recalls and most active work I try to use a toy as much as possible. I certainly don't believe that all agility dogs must tug, but with a dog who values toy play I feel that it has the potential to create a higher level of arousal and intensity that can be hard to match with food. For many dogs, a ball can create even more intensity and speed than tugging can. I use kibble during shaping games or during parts of class where Zumi is going back and forth between me and a helper. I love using food for a more thinking dog and of course the ability to get in many repetitions in a short time is very easy with food.
Definitely switching between food, toys, and personal play can be a skill that needs to be developed. Many dogs learn to expect a certain reward in a certain place, time, or activity. Perhaps agility usually means a tug toy and obedience means food. Or training inside is often a mix of food and personal play and training outside is with a ball launcher. Trying to use a different reward than your dog expects, even if it's a reward they normally really like, can be difficult. From their perspective it's not just switching to a possibly lower value reward but it can be actually punishing if they see it as a loss from what they wanted.
I am currently using kibble as a reward for her in class, versus switching to string cheese or other higher valued food rewards, because it works. I am very happy with the amount of engagement and intensity I am getting from it! The only downside I see in using kibble is that it can be hard to see on the dark colored mats for thrown rewards. Switching to a higher valued food reward would give me plenty of options to help with visibility but comes with it's own downsides.
Mainly in that once a higher valued reward is out there, Zumi's food love isn't strong enough to switch back to a lower valued food. The Corgi and the labs I've raised haven't cared at all. I'm pretty sure they would be just as excited about pocket lint! But Zumi wanted nothing to do with me after the instructor decided to switch to string cheese during our turn one week and thus I needed to grab some too. Even after that exercise and a little bit of time in her crate during a break, Zumi took several minutes before she would work for kibble again eagerly. I'm sure if I wanted to use a toy after that it would have been impossible seeing that I'm still trying to work on kibble to tug with her. In general I try and use toy play in the beginning of class while she is most excited about it. Before she has an opportunity to earn food and "forgets" how much she loves to tug!
On a similar note, I think it's important to remember that you can't use a reward your dog doesn't love! Personal play (no food, no toys) is not a high enough value yet to be of use with Zumi. We are still working on developing games she likes and ways of playing that don't involve lots of teeth! This will be way more important in obedience than in agility. But I see a lot of people who insist that their dog play tug with them when their dog has zero interest. Perhaps their dog loves to do it at home, perhaps not. Working on building a motivator has to be done in an environment where the dog is relaxed. And the motivator can not be used as a reward if your dog doesn't want it! Even with Zumi right now who loves to tug, I'm not yet asking her to tug in class as a reward for a behavior once I bring out food (before the food it's a non issue and is a reward!). I do occasionally attempt some toy play separate from everything else. More of play with me because it's fun versus play because you earned it.