Those that are unregenerate can not choose Him. All men do as they will to do (that is free will) .
Let me ask you a question, can you honestly say that having God give you a will that can and will choose Him is an evil thing? An undesirable thing?
When I hear people say things like " God does not FORCE you" I wonder if we live in the same reality . Is it an undesirable fate to be in the presence of the Eternal God? Is it a fate worse than death to run to Him ? Why would He have to FORCE us to come?
Did Lazarus yell out from the tomb, "I do not choose to live thank you anyway" .
We read in Romans that every man will stand before God without an excuse .That is a very important principle . God reveals himself to all men , He takes nothing away from from men,He refuses no one that comes to Him. All men are held responsible to respond to the call , and will be judged so. Paul tells us that no man seeks God or even understands God. When God regenerates (quickens) us , our ears and eyes are opened and we can for the first time see God without the barrier of Adams sin separating us from Him. We then will seek and understand.
As Calvinists , we do not see God as taking anything away from the reprobates, but as adding to the elect the grace to see and hear without the static of sin. He does not remove from us our free will, He makes it more free in Him.
Undesireable? No of course not. But there are lots of things that are desirable that also do not exist.
Can you say God giving you a hug is an evil thing? Does that mean people get hugs from Him every day?
Your points boil down to "If everyone had the choice - how could any of them possibly say no???" Which is no different that asking how Adam and Eve could have screwed up when they had everything so perfect. Free will combined with " the sin that is within me that is not my sin" makes it very possible. Not "desirable", but far too common. We are bent.