"Taking 2 lessons a week at $100 per lesson is $800 per month. 3 lessons a week costs $1200 per month. This comes out to $9,600 or $14,400 per year."
And you get to divide that by two.
"One needs a sponsor, trust fund, or money to pay for all of this."
No - one needs a job and a willingness to sacrifice other things to achieve your goals. For most amateur succesful competitors today, that job is teaching dance... which really doesn't pay very well!
"Still there is no guarantee that you will win anything."
Exactly. Pouring money down the drain won't solve your problems if you do not have the time, determination, and ability. If you do, there seems ample evidence that you can succede while supporting yourself.