[0] https://www.helgilibrary.com/indicators/rice-consumption-per...
[1] https://en.wikipedia.org/wiki/List_of_countries_by_obesity_r...
Also, a note: not everyone can eat a meat-heavy diet. I cannot. I don't have a gall bladder - like many others - and cannot digest the fats well. Eggs are too much sometimes.
Not only that, but you can eat perfectly healthy without actual meat. Low quality foods - meatless or not - generally have a lot of empty calories and aren't very filling, making it easier to gain weight.
I would argue that the answer is the nutrient to calorie ratio. Many nutrients for few calories is good, but few nutrients for many calories is bad. By this analysis, plant-based food is "low quality" and animal-based food is "high quality".
So if calories are what make you gain weight then it makes sense that people following a plant-based diet would gain weight compared to people following an animal-based diet because they will need to consume more calories to acquire the same amount of nutrients.
The main reason folks lose weight on meat-heavy diets is because they tend to eat less and therefore eat fewer calories. But there is more than one path to eating fewer calories. Lentils and beans tend and some vegetables tend to be rich in fiber, which keep you full longer and tend to mean you eat less. Not to mention that a plate full of vegetables and beans tends to have fewer calories than a plate of meat - and hold a variety of nutrients as well.
You are correct when you state at the beginning of your second paragraph that people on meat-heavy diets tend to eat fewer calories. But you are incorrect when you state at the end of that paragraph that a plate full of plant food will have fewer calories than a plate of meat. If this were true, then people on meat-heavy diets must not be finishing their plates of food, since they end up consuming fewer calories.
As for your own diet, it's a good thing you have the eggs and cheese in there! That's where your vital nutrition is coming from. The rice and beans and veggies are just keeping you alive with calories but not carrying their weight in terms of nutrition. You could be doing worse, such as eating a bunch of bread and pasta, or even worse than that, soda and candy, but you could also be doing much better by eating mostly steak and sardines.