Okay, with Henry Cabot Lodge, I'm not sure when his earliest plausible shot might have been, but perhaps the 1890s, because he we proposing serious national legislation by then?
In any case, whether nominated any time from the 1890s through 1920s, I think hallmarks of his policy agenda would be support for US imperial foreign policy overseas and a strong navy and ability to intervene in Asia-Pacific and Latin America, immigration restrictionism, since he seemed by the turn of the 20th century to have a prejudice against the 'new immigration' from southern and Eastern Europe, and, especially in the earlier parts of his national career, an opposition to the disenfranchisement of blacks, as shown by his eponymous Lodge Bill, which would have reduced southern representation in line with the proportion of adults actually enfranchised as voters.
Leonard Wood was not discussed as a Presidential candidate until 1920. His concerns would have been about 100% Americanism, immigration restrictionism, physical fitness, other macho Theodore Roosevelt type stuff, likely would have continued to be more uptight about the Red Scare than Harding was. While Congress wasn't going to keep the purse open for unlimited Navy budgets, which would compel participation in the Washington Naval talks, he may balk at US Treaty commitments to not fortify US positions in the western Pacific like the Philippines and Guam.
