@GentleOx the answer to this excellent, opening, question is "it depends".
Strictly speaking every type of exercise builds muscle and burns fat. But they don't all do it in the exact same way or at the exact same rate so grading one as superior to another is predicated on context and the context is dependent on you. Your fitness levels, experience with exercise and goals when exercising all play a role so the one that will be superior, for you, to the others will depend on why you're exercising, how long have you been exercising for, what your expectations are, your particular lifestyle and what you hope to accomplish through exercise.
If your goal, for instance, is to get stronger then building muscle is a byproduct of that process and the choices you will make in terms of exercise. If, on the other hand, you want to build larger muscles or put on more muscle specifically, the protocols you need to put in place in your training should all be aligned to that goal.
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2018 study that compared muscle thickness and 1 repetition maximum (1RM) in two groups where one used push-ups with a progressive load in execution, and the other used the classic benchpress, also with a progressive load in added weights, found no difference in results between the two. So, to the extent on developing some specific muscle groups at least, traditional resistance training and calisthenics are the same.
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2023 study looked at the effects of isometric exercises to failure compared to a combination of isometric exercises
and resistance training. In this particular study they were measuring for muscle size increase (hypertrophy) and "maximal voluntary isometric contraction (MVC)" i.e. joint strength (as the focus was on the elbow joint). What they found was that there was no difference in strength between the two control groups so those who did isometric exercises to failure developed the same strength as those who did traditional resistance training
and isometric exercises to failure but the second group did experience more hypertrophy than the first.
Both these studies are counterintuitive which shows the many complex pathways that are activated by exercise which then lead to strength. All of which brings us to my opening statement of "it depends". Your goals and particular set of circumstances (and I am using the world "you" generically here) will determine the particular training combination you will want to put in place.
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2019 study showed that when it comes to developing joint strength, isometric exercises were superior to dynamic ones (i.e. in sports that involve jumping and kicking for example) resulting in fewer injuries and greater increases in performance, faster. Again, this shows the need for specificity in our fitness goals. We know, for example, that dynamic performance (i.e. moving the body in a complex way through three dimensional space) requires equally complex neuromuscular adaptations that control both force (maximal muscle velocity) and precision. This study highlighted the effectiveness of isometric training in achieving this precise combination of effects, particularly when an athlete had mobility issues because of injury and could not perform a dynamic movement for that reason. It also showed that with a change in protocol isometric training could also induce hypertrophy when required.
In their totality this trio of studies show that when it comes to muscle size and strength you need to be fully aware of what you are training for and work, intentionally, to create the best combination for yourself.
The second half of your question had to do with fat loss. All exercise involves an increase in energy expenditure and that increase in energy expenditure, over time, with most other factors remaining the same, will deliver a decrease in the body's stores of fat. A
2021 study that used young women as its subject, looked at both isometric and isotonic (i.e. resistance training and dynamic movements) exercises and concluded that they both lead to a decrease in stored fat loss with the dynamic exercises having a slight edge.
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2022 study that used 100 college-age students as its subjects also found similar beneficial results. Both isometric and isotonic exercise helped decrease the body's store of fat (and again isotonic exercises had a slight edge). Both isotonic and isometric exercise helped increase strength and gain muscle. However the study also showed that when it came to increasing cardiovascular and aerobic performance, surprisingly, isometric exercise delivered better results. This is surprising because the respiratory load (i.e. the rate at which we breathe when exercising) would, from an experiential point of view, be heavier when we are lifting weights than when we hold a static loaded position. My guess here (and it is only a guess) is that the improvement came not so much from any respiratory adaptation but from a significant improvement of joint strength and joint stability which, in this case, made the aerobic exercise the students were tested on, easier for those those who did isometric exercises than those who engaged in isotonic ones (i.e. resistance training and/or dynamic training).
This same study also showed that isotonic training, i.e. resistance training, again somewhat counterintuitively, lead to a greater range of motion (ROM) and better flexibility than isometric training. The reason I say somewhat counterintuitively is because the general perception is that resistance training leads to muscle and joint stiffness and loss of flexibility.
So you can see there are trade-offs. No method of exercising is exactly the same as another. Fat-loss in the studies cited, for example, was always a little higher with resistance training than it was with isometric training. This is to be expected when we realize that fat loss, really, is carbon dioxide molecules (CO2) expelled from the body with each breath we take and we tend to breathe deeper and heavier when lifting weights or performing a dynamic movement than when we hold a static pose. Note, however, that this fact did not translate into improved cardiovascular performance for the students in the 2022 study, mainly I suspect, for the reasons I've already mentioned above.
I hope all this helps shed some light on what is, admittedly, a fairly complex subject.