Stretching is typically a slow, drawn out process. You gradually get more and more limber day after day, month after month. You will not benefit the most by stretching in one position only, thus it is not likely that you will stay in the supported splits position until you finally reach the floor with your chest flat on the ground.
Stretch in this position for a while (center splits with both legs to the side), then change to single leg side-hip stretches, front-hip stretches (and do the front splits with left leg forward, then right leg forward). Also sit on the floor with your legs wide apart, and begin to slowly bend at the pelvic region (keeping your spine straight), and bring your chest and chin towards the floor in front of you, and to each leg on the left and right.
This is where you will gradually achieve a wider sitting split, with your body upright, and then lay forward till your chest touches the floor flat, but you might not have your legs at a full 180 degree split yet. In time, the supported split will allow you to gradually drop to the full split position, but it won't matter then if your body is up or down because you will be able to do both. However, once you are out in the full splits with your chest on the floor, it will be one thing to "push" yourself to an upright postion (no problem), and another thing to "lift" your torso up with the muscles of the back, abdomen, hips, and legs without using your hands. That will take some practice.
In answer to your question about supporting your weight, you should do it both ways. There are three main goals to stretching. One is to gain flexibility to increase range of movement. Supported stretches take the weight off your muscles and allow them to relax so they will not resist the stretch yet using gravity to your advantage. You will gain more flexibility, at a good pace, if you gradually adjust the amount of your own body weight you allow to force your hips to the floor.
The second goal to stretching is to gain strength in your legs while you are increasing the flexibility. Therefore, you should spend some time on stretching in the splits without supporting your weight, and some time while standing on one leg with the other held in the air, or with your foot supported on an elevated object (a stable one that won't fall over).
The third goal in stretching is to gain agility. This means that you should strive for mobility of the hips while in a stretched position. Your hip sockets don't just flex in one direction. Once your legs (in single-leg or double-leg stretches) are fully extended, move the hip positions around until you gradually shift from minor adjustments forward or back to a complete change from side-hip to front hip and back again.
Supported stretches help flexibility, and unsupported stretches promote strength of muscles and maintain stronger hips socket control, but tends to work against the flexibility phase. Don't over-stretch and get loose, weak muscles. You need both supported and unsupported stretches to balance out the objectives.
I hope this helps.
CM D.J. Eisenhart