I an a retired band teacher who taught beginners to play all of the band instruments for over 27 years. As such I really like to offer assistance to players starting out. Please don't take offense if I offer some constructive suggestions on tone quality and control of the sound.
1. Your embouchure is not yet firm enough to produce a "characteristic" saxophone sound.
There are two good ways to check the tightness of the embouchure muscles. One way is to play low A and then flick the neck octave open with your free hand. If the high A sounds flat the embouchure is too loose. If the note stays on high A before dropping down, the embouchure is too tight. If the note pops to a nice sounding and in tune high A momentarily and then drops back down, the embouchure is set at the correct tightness.
The other way is to play the mouthpiece and neck apart from the saxophone. The ideal input pitch of this "small saxophone" or "tone producer" as I call it is Ab Concert. The embouchure setting that produces that pitch produces a nice sound when accompanied by lots of air, and "locks in" the best intonation your sax is capable of producing.
There are several different styles of saxophone embouchures that work. The one I have found to be the best when starting beginning players is the Larry Teal ring of muscles around the mouthpiece. To do this one says "EE" "OO" with the muscles around of the mouth both pushing in and pulling out in opposition to one another making a tug-o-war. On saxophone the "OO" muscles win, but the "EE" muscles keep pulling out at the same time. The embouchure should feel like a big rubber band around the mouthpiece.
2. Your reed sounds too soft.
This is related to #1 since too soft a reed does not require a well developed embouchure to make a sound. Oftentimes when players try to play too soft a reed on saxophone or clarinet with the correct embouchure the reed closes off. For most mouthpieces suitable for students a #2 or #2 1/2 Rico or its equivalent is a good place to start. The often heard advice to start on a #1 1/2 because it is easy to make a sound is not a good idea in my experience. All a very soft reed does is encourage poor playing habits and delays the development of good muscle tone in the embouchure.
Besides listening for a "flabby" sound in the upper octave, beginning players can try the "pop" test to check the strength of the reed. To do this test you take the mouthpiece off, close the open end with your finger, and suck all the air out of the mouthpiece. When all the air is out, quickly pull the mouthpiece from the mouth, and monitor the time it takes for the reed to pop open. If it pops in a second or less, the reed has sufficient strength. If the reed takes over 1 second to pop, it is too soft.
3. Make sure you are placing the top teeth on the mouthpiece.
Some advanced players like to use what is called a "double lip" embouchure. This is not very common on saxophone, and is not a good idea for beginning players. On the alto the top teeth should press down about 5/8" in back of the tip. If it is uncomfortable to set the top teeth on the mouthpiece and play, I recommend getting a thick mouthpiece patch. It will not only make playing more comfortable, it will help your consistency by guiding your teeth to the same spot each time you play.
4. Let the neckstrap hold the weight of the saxophone.
From the video it looks as if you might be lifting the weight of the saxophone slightly when you play. Let the saxophone hang by the neckstrap and using your thumbs balance the sax in front of you. Keeping the head perfectly erect, see where the tip of the mouthpiece touches on your face when you tilt the sax toward your mouth. It should just touch the curve above your chin. To play just tilt the head downward slightly and the mouthpiece will go into the mouth at the correct angle.
5. Keep blowing lots of air when you play.
This is really the best part of the practice demonstrated on the video. Most players starting out don't use enough air. You are doing this quite well and should be complemented. The low notes coming out with a big full sound is evidence that you are blowing lots of air. The next step is to learn to control the sound using that much air which will give you a beautiful tone. Playing long tones is the most valuable exercise one can do when learning the saxophone.
I like to have my students play their mouthpiece and neck watching the second hand on a clock. They try to hold the tone 15, 20, 25, 30 seconds with a full tone and well controlled sound. Then they do the same exercise on the saxophone playing top line F (Ab concert) which sounds the same as the mouthpiece and neck. The better the mouthpiece and neck alone sounds, the better the whole saxophone will sound as this long tone exercise is practiced.
Good luck in your musical endeavors. Keep us apprised of your progress. It takes a lot of courage to do what you have done. Bravo for that.