Where did the idea come from that support for a second New Deal means support for Jim Crow?

[Jim Lane]

16. That process of change also included Truman, Humphrey, and Kennedy

It was Truman who desegregated the armed forces in 1948. He was pushed further on the issue when the civil rights forces, with a young Hubert Humphrey as their leader, secured a strong civil rights plank in the 1948 Democratic platform, spurring the Dixiecrat walkout.

The party's support for civil rights caused the erosion and then collapse of what had been the "Solid South":

In the 1960 election, the Democratic nominee, John F. Kennedy, continued his party's tradition of selecting a Southerner as the vice presidential candidate (in this case, Senator Lyndon B. Johnson of Texas). Kennedy and Johnson, however, both supported civil rights. In October 1960, when Martin Luther King Jr. was arrested at a peaceful sit-in in Atlanta, Georgia, Kennedy placed a sympathetic phone call to King's wife, Coretta Scott King, and Robert Kennedy helped secure King's release. King expressed his appreciation for these calls. Although King made no endorsement, his father, who had previously endorsed Republican Richard Nixon, switched his support to Kennedy.

Because of these and other events, the Democrats lost ground with white voters in the South, as those same voters increasingly lost control over what was once a whites-only Democratic Party in much of the South. The 1960 election was the first in which a Republican presidential candidate received electoral votes in the South while losing nationally. Nixon carried Virginia, Tennessee, and Florida. Though Kennedy also won Alabama and Mississippi, slates of unpledged electors, representing Democratic segregationists, would award those states’ electoral votes to Harry Byrd.

The parties' positions on civil rights continued to evolve in the run up to the 1964 election. The Democratic candidate, Johnson, who had become president after Kennedy's assassination, spared no effort to win passage of a strong Civil Rights Act of 1964. After signing the landmark legislation, Johnson said to his aide, Bill Moyers: “I think we just delivered the South to the Republican Party for a long time to come.”[30]

In fact, although the OP is correct in generalizing that the New Deal left racial discrimination in place, even FDR played a role in the Democratic Party's shift. He signed an executive order "banning discriminatory employment practices by Federal agencies and all unions and companies engaged in war-related work," and he created the Fair Employment Practice Committee to enforce the order.

I'm absolutely not belittling LBJ's role in signing the Civil Rights Act, along with the Voting Rights Act and the Fair Housing Act. Nevertheless, other Democratic leaders deserve to share in the credit for the change.