Your last paragraph is exactly right, and is exactly the point that the verse makes, and that RC made. Those drawn will be lifted up.John 6:44 No man can come to me, except the Father which hath sent me draw him: and I will raise him up at the last day. (KJV)
This verse, IMO, does not support Universalism which teaches that all people will eventually be saved through the universal redemption of Jesus. Therefore, when someone quotes this verse in reference to the drawing ministry of God the Father, it does not make them a Universalist.
This verse is similar to Hebrews 9:22 where the author tells us, "...without shedding of blood is no remission."(KJV) The blood has been shed; there can be no doubt, and that, again IMO, is the "drawing element" mentioned in John 6:44.
God has given us Jesus Christ and His shed blood, all those who are drawn to Christ as Saviour will be lifted up as God has promised.
But the problem is that it also affirms the opposite: those not lifted up were not drawn. The blood shed (that we agree on) was not a "drawing element." Heb 9:22 has no similarity whatsoever to this verse.
This verse draws a circle around those drawn and those lifted up. They are the same. If you believe that all men without exception are drawn, then you must also believe that all men without exception are raised up, or else deny the teaching of the verse.