It may not be possible (or you may not want) veg/salad at every meal - e.g. breakfast - but yes, you certainly should be eating fat with every meal. It's not about speeding up weight loss, it's about getting the right proportion of nutrients. We should be eating fat:protein:carbs.
Obviously with eggs and cheese you are getting fat as well, but with eggs there is more protein than fat. With meat, you are getting some fat, but this obviously depends on the cut of meat - chances are you're actually eating more protein.
Fat does not provoke an insulin release, whereas protein does (albeit a lot less than carbohydrate).
You should certain not be eating meals that are just fat or just protein. Aim to combine the three nutrients for best effect and health.
If your weight loss is slow, you might also be wise considering reducing the amount of tea/coffee you're drinking. Some believe that caffeine also spikes your insulin.