To answer the question of which cardio machine is the best for weight loss, I'm going to train you all to be lawyers (and for a lot less time and money than you would spend for law school). The answer to this, and almost all legal questions is, "it all depends." This is a universally true and a universally frustrating answer. But as with many things in law and life, the answer is mostly subjective. Sure there are the heavy hitters like treadmills, rowers, ellipticals, and stationary bikes. But the key is, which of these will you use? Regularly. And push yourself on? A machine is only as valuable as you make it.
If you get on a treadmill and get bored after five minutes (e.g. yours truly), it will not be the weight-loss machine of your dreams. But if you spend treadmill time willingly and find variety through things like hill, interval, and run/walk workouts, it may be the ticket to burning the calories you need to achieve the weight loss you want. Walking and running burn up to 400 and 600 calories an hour respectively. But those calories won't burn if you don't hop on, or if you use your home treadmill as a clothing rack.
Rowers are total-body cardio machines and can really boost your fat-burning potential. Unlike the treadmill that you turn on and start moving, there is a little bit of technique to a rower. Once you get that down, you can do any combination of power and endurance work to get your heart rate up and your energy consumption rolling. Most gyms have a rower available, and they are available for in-home use too (stand them upright after using and they take up less room than a treadmill or elliptical — bonus!). If you've never used one before, have someone demonstrate (or watch a video online), and give it a try. Trying something unexpected may increase its appeal as a workout tool, and you'll want to do it more often because of that. And if you really get going, you can burn over 800 calories an hour. Bear in mind that an hour is a long time rowing with your entire body, so plan accordingly.
Ellipticals are a standard gym resource for a reason. They are low impact and produce good calorie expenditure. Because it is only indirectly weight bearing, unlike the treadmill, the demand on your body can be less. And we need an energy deficit in order to trigger the body to seek sources in the body, specifically fat cells. You can burn over 600 calories an hour on an elliptical if your effort is sufficient. If you are easily distracted by a screen and your favourite show, don't have them on during your workout. You need to focus to ensure your intensity doesn't waver. That will be the key to your long-term weight goals.
Indoor cycles are great fitness tools and, like ellipitcals, are low impact and easy to adjust to individual users. If you don't like classes, most gyms provide a few stand-alone cycles, or you can join an online service and do the workouts from home with your own equipment. Cycling can utilise over 600 calories an hour, depending on your power output. Most of us learned to ride a bike when we were kids, and there can be a childlike joy to getting back on a bike, if you have the right combination of direction and music. There is some physical adjustment time required (the "saddle sore" issue is real, but it does go away), but otherwise, it is a lifetime exercise for almost anyone.
The problem with answering "which is the best machine" is that your answer is going to be different than mine, based on our levels of fitness, our experience with different modalities, and the appeal of the actual work. You may be a treadmill junkie, and I may consider it a torture device. I may live for cycle class, and you can't get over being sore in your undercarriage. You may look forward to your checkout time on the elliptical with the evening news, and I may look longingly at a rower and pretend that I'm heading to a new life in Fiji. All of these are valid and right.
As much as a definitive answer might make someone feel better about what they are doing, only your experience and your ability to achieve the goals you are setting will tell you if this is the right machine for you. If you have hit a plateau, try adding in a different machine for some cross-training. You don't have to break up with your favourite, but swipe right on something else once in a while. Your body and weight-loss journey will appreciate it! Case dismissed!