It could also mean, reasonably, that no one calls Christ Lord but by the Spirit just as it says.
I just showed two ways the verse can be understood, just as you stated, without proving one must first be regenerated before believing.
Such a theology, however, is refuted by scripture
John 12:40 (KJV 1900) — 40 He hath blinded their eyes, and hardened their heart; that they should not see with their eyes, nor understand with their heart, and be converted, and I should heal them.
God blinded men to prevent their belief. Why would God blind someone who had no ability to see?
Why prevent from believing those who had no capacity for belief?
All he would have to do is not regenerate them if your theology were true.
Luke notes had men not grown hardened they could have believed
Acts 28:27 (KJV 1900) — 27 For the heart of this people is waxed gross, and their ears are dull of hearing, and their eyes have they closed; lest they should see with their eyes, and hear with their ears, and understand with their heart, and should be converted, and I should heal them.
This verse would be untrue if regeneration was first required
unregenerate men are shown to receive the word with joy and believe
Luke 8:13 (KJV 1900) — 13 They on the rock are they, which, when they hear, receive the word with joy; and these have no root, which for a while believe, and in time of temptation fall away.
Are you going to claim God's regeneration is temporary?