I've moved my old answer to the bottom, as it is less relevant than the update.
As per the comments, an elaboration the idea behind this and how it relates to your question:
As I understand it, Iluvatar granted the gift of death to Men, thus making them different from (immortal) Elves.
It's important to consider that mortality is the gift (to Men), not immortality (to Elves). If you ask any (out of universe) human, they would immediately assume that immortality is the gift here, as everyone wants to not die. This is not the case for Tolkien; and I believe this is intentional, so as to stress the importance of life.
Life is considerably more important if it can be taken away by death. It gives meaning to not being dead (yet). A bit more pedantically, something that cannot die cannot really be described to be alive ni the first place, by semantical definition of what "life" is.
Therefore, limiting the lifespan of Men is an intentional goal. It is the method through which Men obtain the virtue to shape their (limited) life.
The text you quoted somewhat refers to this virtue being born from risk:
but they should have a virtue to shape their life, amid the powers and chances of the world, ...
The emphasized part, especially "chances", seems to imply the existence of the possibility that everything can be taken away. "the chances of life" can be argued to be synonymous with "the possibilities of dying".
But why is having a limited lifespan necessary to want to shape your life?
Because you can't put things off. Elves could, if they wanted, take a 2000-year-sabbatical before they do whatever it is they want to do (= shaping their life). But Men are not afforded that luxury. They have limited time, and they must therefore make the most of it.
I'll refer to an out of universe analogy, American Idol (didn't see that coming, did you?)
During the initial selection phase, contestants are limited to one minute. They are given a limited time to show their worth. Why is that?
So that they forgo the unnecessary and get to the point about showcasing their skills. Their limited time ensures that they won't waste everyone's time with a boring setup, and instead skip to the good parts.
I believe the same behavioral manipulation is at play for Men. Having a short lifespan dramatically increases the need to make sure that you do the important things before anything else. Therefore, logically speaking, Men must have a larger desire to quickly (relative to an Elf's lifespan) achieve greatness.
To directly answer your question
I don't think this is a case of free will, as your question assumes. I think this is a matter of stressing the importance of life, the consequence of which is an increased desire to live life to the fullest.
To transcribe my interpretation over the quote you referenced:
Therefore he willed
Therefore he manipulated the system
that the hearts of Men should [seek beyond the world] and should [find no rest] therein
that Men will carry around a [burning] and [incessant] desire
(from the sentence structure, this is implied to be a negative effect)
but they should have a virtue to shape their life,
but that they also carry a passion to truly live their life to the fullest
(due to the but, this is implied to be positive, compared to the earlier negatives)
amid the powers and chances of the world,
in a world where their life could so easily be swept away
beyond the Music of the Ainur, which is as fate to all things else
which is different from the Music of the Ainur, fate, which inherently makes a person's actions meaningless since everything is predestined already.
("Fate" implies that there is no reason for Elves to want to shape the world, as they consider everything to be set in stone already)
My initial answer
You can't teach an old dog new tricks.
It's a very often recurring pattern where old men are less eager to change their ways than young men. When considering Elves and Men, Elves are the old men, and Men are naive younglings.
Elves are immortal. Therefore, they are considerably wiser than Men, who live short lives and are considerably more naive than Elves. It's a matter of having lived less long, and therefore having learned less life lessons.
The older you are, the less inclined you are to try something new. If you've seen it all before, you've lost your childlike wonder and will act based on past experience, without a need to experiment with new things (as nothing is new to you anymore).
As Elves are considerably older than Men, they should then be less willing to learn new things that contradict their extended life experience.
Elves do not have "less free will", they simply have more life experience and therefore are less likely to run into something they haven't seen a thousand times over.
Note You see a similar thing between Arwen and Elrond. Even though they are both Elves, Arwen is younger than Elrond, and she is more hopeful, optimistic and naive.
Elrond is considerably more insistent that things will be the same as they were before. His opinions are clear: Aragorn is no better than Isildur, Men will remain fallible as always, there is no point to love a mortal man for a limited period of time when you are immortal. Elrond considers the long term, whereas Arwen considers the short term (as do Men, since their lives are limited to a short term compared to that of an Elf)
I think this trait in character isn't inherent to Elves or Men, but rather the age (and expected lifespan) of the person, regardless of race (though race can obviously make a difference as to expected lifespan).