Are you saying Blacks in the South were not Republicans? I'm pretty sure they were and it would have helped the Republicans if they were allowed to vote. I'll take your word on Teddy's view of formal democracy in the South, however your reasoning of what Blacks voted in the North and South is not accurate. Blacks voted Republican, and would until about the 1940s or 50s.
Blacks in the South were almost uniformly Republican until after WW II. But between 1900 and the 1960s, they couldn't vote. (Note: black voting did
not shut down completely after 1876. There was a long and gallant rear-guard action; black Republicans were elected as U.S. Representatives here and there, right up to 1898, and other blacks won state legislative seats. (I have no records how many and when, but there were some.)
(Also, In Memphis, blacks voted under the control of Boss E. J. Crump, a Democrat. This started no later than the 1920s, and possibly in the 1910s.)
But that wasn't the issue. In Roosevelt's time, the "Tragic Era" version of Reconstruction history had been universally accepted. That is, during Reconstruction, blacks had voted - but being stupid and ignorant, were easily gulled into electing extremely corrupt white "Carpetbaggers" and "Scalawags" to state and local offices; also some corrupt and debauched blacks. Klan terrorism was depicted by Southern writers, and in the Democrat national press, as a struggle to "redeem" the South from this oppression.
There
were some notorious incidents of corruption among the Reconstruction-era Republican officials, also heavily publicized. It is worth noting that in 1872, Horace Greeley, a would-be reformer, bolted the Republican Party in protest over the corruption of the Grant administration, and then accepted the support of the Democrats including the Southern "Redeemers".
By 1900, this was "old news". It was also clear that white Southerners would
never accept black voting again. It was not because of any help it might be to Republicans nationally (black votes could carry at most only South Carolina). It was because blacks were majorities in many counties, and would control local government, including law enforcement. So there wasn't really anything for national Republicans to gain from black voting in the South, and most had accepted the idea that black Republican majorities would be an embarassment, as allegedly in the 1870s.