Does freedom from birth and death mean there is no freedom to experience birth and death without desire?

There is no birth with full awareness of the truth, and

once the truth is known there is no longer a possibility of experiencing the cycle forever.

Does freedom from birth and death mean there is no freedom to experience birth and death?

There's no such a thing as the "freedom" to experience birth and death. If one hasn't attained enlightenment, one'd have no choice but to experience birth and death! It's like as long as there's still oil in an oil lamp, the fire will continue to burn, or at least the potential to burn continue to exist nevertheless. But once the oil's been completely drained/exhausted, there's nothing left there for the fire to consume and carry on its existence. Hence the freedom from birth and death means one's broken free from its chains for good, not that one is free to break the chains and then put it back on again from time to time if you feels like doing. It's like saying the freedom from lust is to take a break from it, but then one's free to re-engage some hoochie coochie from time to time if s/he feels like it!

Existence is suffering. Cessation of existence is a realization of the truth of the cessation of suffering.

“Just as, mendicants, even a tiny bit of fecal matter still stinks, so too I don’t approve of even a tiny bit of continued existence, not even as long as a finger snap.” AN1.329

“One perception arose and another perception ceased in me: ‘The cessation of existence is nibbāna; the cessation of existence is nibbāna.’ Just as, when a fire of twigs is burning, one flame arises and another flame ceases, so one perception arose and another perception ceased in me: ‘The cessation of existence is nibbāna; the cessation of existence is nibbāna.’ On that occasion, friend, I was percipient: ‘The cessation of existence is nibbāna.’” AN10.7

When you say;

the cycle of birth and death produces suffering but ,once the truth is realized suffering has no power ,so living in this world shouldn't be any longer a problem.

You are slipping by the definition of suffering as all conditioned phenomena;

Monks, there are these three kinds of suffering.[1] What three? Suffering caused by pain,[2] suffering caused by the formations (or conditioned existence),[3] suffering due to change.[4] It is for the full comprehension, clear understanding, ending and abandonment of these three forms of suffering that the Noble Eightfold Path is to be cultivated..." SN45.165

Even the existence of what can be grasped with wrong view to be an Arahant or personal and belonging to an Arahant, that is classed as suffering.

The Arahants are disenchanted with suffering and thus with the breakup of the body do not take up another body and their consciousness does not become established in any sphere of existence, they abandon all being;

These two Nibbāna-elements were made known By the Seeing One, stable and unattached: One is the element seen here and now With residue, but with the cord of being destroyed; The other, having no residue for the future, Is that wherein all modes of being utterly cease.

Having understood the unconditioned state, Released in mind with the cord of being destroyed, They have attained to the Dhamma-essence. Delighting in the destruction (of craving), Those stable ones have abandoned all being. Iti44

Existence is a truth and it's cessation is also a true principle to be realized and when known as it actually is, it is known to be the highest bliss also called Asoka [Sorrowless] state.

It's hard to see the unaffected, for the truth is not easily seen. Craving is pierced in one who knows; For one who sees, there is nothing. Ud8.2

Cessation of the conditioned depends on the unconditioned element, if there was no unconditioned then the cessation principle wouldn't be possible.

There is that dimension, monks, where there is neither earth, nor water, nor fire, nor wind; neither dimension of the infinitude of space, nor dimension of the infinitude of consciousness, nor dimension of nothingness, nor dimension of neither perception nor non-perception; neither this world, nor the next world, nor sun, nor moon. And there, I say, there is neither coming, nor going, nor staying; neither passing away nor arising: unestablished,[1] unevolving, without support .[2] This, just this, is the end of stress. Ud8.1

There is, monks, an unborn — unbecome — unmade — unfabricated. If there were not that unborn — unbecome — unmade — unfabricated, there would not be the case that escape from the born — become — made — fabricated would be discerned. But precisely because there is an unborn — unbecome — unmade — unfabricated, escape from the born — become — made — fabricated is discerned. [Ud8.3] (https://www.accesstoinsight.org/tipitaka/kn/ud/ud.8.03.than.html)

When cessation principle is realized it is known as pleasant and any existence in the light of it is categorically known as suffering.

It is as if a tired person was offered a seat just to have that seat pulled out from underneath him, not knowing better, might think; 'This good person is offering me a seat out of kindness, why don't i accept?'. So he accepts, time and again, not understanding as it actually is, that it is out of cruelty and for the purpose of suffering that he is offered a seat.

If later the tired person the was offered a stable relief then he would discern the cruelty in what was previously offered and if offered to take a seat would pass on the offer, discerning the deal as suffering.

Therefore the mind of one who has realized the cessation principle realizes the truth of suffering, the truth of arising of suffering, the truth of it's cessation & the path to the cessation; and thus inclines to Nibbana which is the utter extinguishment of the conditioned & of the causes which beget the conditioned existence.

There he said to the monks, "This Unbinding is pleasant, friends. This Unbinding is pleasant." When this was said, Ven. Udayin said to Ven. Sariputta, "But what is the pleasure here, my friend, where there is nothing felt?" "Just that is the pleasure here, my friend: where there is nothing felt. AN9.34

Now it's possible, Ananda, that some wanderers of other persuasions might say, 'Gotama the contemplative speaks of the cessation of perception & feeling and yet describes it as pleasure. What is this? How can this be?' When they say that, they are to be told, 'It's not the case, friends, that the Blessed One describes only pleasant feeling as included under pleasure. Wherever pleasure is found, in whatever terms, the Blessed One describes it as pleasure.'" MN59