Most of the criticism was over FDR cashing in undeservedly on Teddy's popularity and good name. During the 1920 campaing, when FDR was candidate for Vice-President, the Chicago Tribune ran an editorial criticizing FDR, saying, "he probably would not have been thought of" for the Democratic ticket if it weren't for his last name, and that "Franklin is as much like Theodore as a clam is like a bear cat."

Teddy's son, Theodore Roosevelt Jr., campaigned for Harding against the Cox/FDR ticket that year. Cox lost and FDR never served as VP.

FDR got his revenge four years later, however, when he campaigned against Teddy Jr.'s bid for the governorship of New York in 1924. Teddy Jr. lost and four years later, FDR ran for the same seat, and won. (The 1924 winner, Al Smith, was running for president in 1928 so didn't run for re-election.) Once again, Teddy Jr. campaigned against FDR.

By the 1932 campaign, the two families had stopped talking to each other. And during that campaign, for the third time, Teddy Jr. supported FDR's Republican opponent. Teddy's widow similarly announced her support of Hoover. Both she and Teddy's oldest daughter, Alice, wrote out statements detailing the relationship between FDR and Teddy, i.e., that there wasn't much of one.

To that point, quite often, when the relationship between the two was written about, it was explained that the pair were fifth cousins and that it was a rather distant blood relationship. An NEA article that ran across the country in the days after the 1932 election even printed a family tree.

So, no, there wasn't much concern over an "elite" family winning the presidency twice. In fact, one of the most in depth AP articles from the 1932 campaign on the Teddy/Franklin relationship pointed out that the situation had happened a couple of times before:

"Only two families have sent more than one member to the Presidency of the United States, the Adamses and the Harrisons. If Franklin D. Roosevelt is elected, the Roosevelts will be the third..."

TL;DR: The issue was only infrequently written about in newspapers around the country, and when it was, it was in terms of Franklin Roosevelt cashing in on a name disingenuously, a name that Teddy Roosevelt, from a rival party, had made popular.