If Smith is “driven” by anything, I think it’s in the sentence from his interrogation of Morpheus M1.
“It’s the smell. I feel saturated by it.” “I must get inside Zion. I must get free.” (From memory).
Smith HATES The Matrix. He wants it to end. Now, being the virus capable of bringing about the cataclysmic system crash of the Matrix that is exactly what he’s trying to do. THAT is why he was waiting in the corridor in the skyscraper keeping the Source and the Architect.
If he could prevent Neo from entering the Source, Smith would have the means to once and for all destroy the Matrix. He already got inside Zion through Bane.