What faults did the Republicans see with presidential Reconstruction?
Basically, the Republicans in Congress believed that presidential Reconstruction was too lenient. They felt that Lincoln's plan made it too easy for the Southern states to return to the Union after having seceded. They felt that this was proven when, under Johnson, the Southern states instituted the black codes.
We must, however, be clear on the fact that the Republicans were not united. There was a moderate faction of Republicans who essentially supported Lincoln's lenient plan. At the same time, there were the radicals who wanted to punish the South harshly. They held that the states had "committed suicide" by seceding and that the states had no right to return to the Union. Therefore, if they were to return, they would have to do so under any conditions that Congress imposed.
The major fault that the radicals, at least, saw with presidential Reconstruction was that it was too lenient to the South.