I think you might be mistaken. The sources I looked up all say that Congress must pass an enabling act, which has to be approved by the President, and the act of admission, like all other acts of Congress also has to be approved by the President. Congress does have the sole power to begin the process.
You may be right. The Constitution is often very general and Congress may have established a practice which does require the President’s signature. I know there have been cases where a President pushed for a state to be admitted—West Virginia (illegally carved out of Virginia) and Nevada (to give the Republicans 3 more electoral votes in 1864—for decades Nevada’s total population was well below the 60,000 which had historically been required before a territory was granted statehood, although it may have briefly been over 60,000 in the 1860s because of the silver rush).