It does.
If anything, the 14th amendment was intended to restrict state power. Congress (Republicans in the North) were afraid that Southern and Northern Dems would undermine their reforms in the South. The 13th Amendment ended slavery, but the Republicans already knew that the Southern states were going to do everything in their power to sidestep the amendment and basically have de facto slavery anyway, so they had to come up with the 14th and 15th amendments to make sure that they had the tools to deal with Southerners with Confederacy sympathies.
The Founding Fathers anticipated it, which is why the wrote Article II, sections 4 and Article 1 section 3, clause 7. The Republicans alive in 1868 knew that there was already a mechanism to deal with a corrupt or rogue president/former president, so I doubt that the 14th Amendment was written with that in mind. The 14th was about controlling Southerners trying to gain seats in Congress.
From Article II
Section 4.
Article 1