There’s a lot in your question. I’m going to relay what my teacher told me recently which is that excellent tone takes a long time, maybe five to fifteen years. No matter what you do, if you practice clarinet in some way every day, and you build up to around an hour of practice or more every day, you will definitely improve in many areas over a period of six months.
One thing is as the student, it’s hard to notice your improvement so you might get frustrated more easily if you don’t record yourself and listen back weeks or months later. Another option is to have a teacher who reminds you of how far you’ve come.
All of that said, your most efficient progress will come by not focusing on only one aspect of playing for long periods (like only tone or only fingering or only articulations). By practicing broadly, all of those things will slowly improve together.
Combining my experience with my teacher’s comments suggests that better tone comes from multiple little areas. Those include proper breathing, strength and endurance of your embouchure, precise control of your embouchure, and even fingering to some extent. Along with embouchure strength comes the ability to play the appropriate reed stiffness. Finally, the quality of the reed and the instrument itself make noticeable contributions.
To summarize my point, there isn’t an exercise that mainly develops tone. The upside is that all of your playing will slowly lead to better tone, especially if you spend some time listening to your tone every time you play and make tiny adjustments to improve it. As your upper range extends and your lips and tongue develop strength, flexibility, and muscle memory of the precise positions, your tone will improve while you work on other things. And it won’t help to be impatient about it - it takes time.
All of that said, there are long tone exercises that can help in several areas at once and long tones are one good type of warmup. My teacher has run me through various long tone exercises in the last six months, each focusing on one or more problem areas that I’ve had.
Many, if not most, teachers will happily give you a lesson over the Internet and won’t mind giving you a single lesson every few months, so you don’t need to spend hundreds a month on weekly lessons to get some professional advice. Consider reaching out on Facebook or online classifieds to find a teacher, who could be anywhere in adjacent time zones to find a teacher who is willing to do a one hour lesson to hear you play a bit and suggest a practice routine for the next several months.