Below are total funds granted by various institutions for need-based, academic/other merit, and non need-based athletic scholarships.
Source: 2015-16 (or most recent available) Common Data Set section H for each school.
Need-Based … Merit … Athletic … School
$286,951,107 $6,961,981 $7,768,892 UCLA
$176,440,595 $5,138,238 $0 Harvard
$169,480,573 $21,650,471 $389,814 TAMU
$158,711,813 $77,017,809 $18,780,386 TOSU
$144,165,756 $7,001,479 $17,036,565 Stanford
$136,295,309 $75,864,987 $19,216,540 Michigan-AnnArbor
$116,369,344 $16,645,416 $13,635,387 Duke
$103,136,000 $2,000,000 $6,000,000 Georgetown
$86,283,301 $10,219,371 $9,290,490 Villanova
$83,059,256 $123,068,758 $10,775,135 Alabama
$58,462,039 $32,163,180 $8,555,412 Nebraska
$47,007,336 $1,041,615 $0 Williams
$38,579,208 $361,249 $2,708,150 Prairie View A&M
So far, I haven’t found any colleges that award more in athletic scholarships than they do in need-based aid.
Some schools do award more in athletic scholarships than they do in academic/other merit aid.
At the Ivies and peer schools (Stanford, MIT etc.) the income ceiling for need-based aid is rather high.
I just ran a net price calculation for MIT, assuming $110K family income, no siblings, no home equity, $55K in financial assets. The estimated net price (after need-based aid) is less than $23K.
Even with a family income of $200K and $100K in cash assets, the estimated need-based grant aid is over $19K and the net price is about $46K.
$200K, or even $110K, is far above the median family income in the USA. About 60 colleges (including MIT) claim to cover 100% of demonstrated financial need. So although I wouldn’t say sports scholarships are a “myth”, I’d say most good students from middle income (or even upper middle income) families seem to have a much better shot at significant need-based aid than they do at big athletic scholarships. Many of those ~60 schools grant virtually no athletic OR merit scholarships (aside from maybe a few specialized endowed scholarships.)