Vyasa Smrithi reads thus
Evam dwijaanitimaapanno vimukto vaanyadoshatah,shruti smriti puraanaanaam bhavedhyayana -
Thus having accomplished ‘dwijata’, the ‘balya krita doshas’ are annuled and he becomes eligible for shruti-smriti-puraana adhyayana.
(Translation VDN Rao)
The second line of vyasa Smrithi (in anvaya(affirmative) ) says
A dvija befits(kshamaha) Shruthi Smrithi Purana adhyayana(study)
This may not mean itihaasapuranaas are inaccessible to non-dvijas ,dvija-bandhus, and sthree(women).They are benifitted through the shravana(listening)of the same, through Pauranikas /Pravachana karthas.
Adhyayana(study) not being their svadharma,(impracticable for them to indulge in it(detailed study) in the light of vedas depend on pauranikas lest may infer the same skewedly)
Study of itihaasapuranaas are critical for those practice veda adhyayana
itihasa puranabhyam vedam samupabrmhayet .
That means that one must study the Itihasas and Puranas, in addition to the Vedas. The reason is to enable a correct interpretation of the Vedas(Mahabharata ,adi parva)
Srimad Bhagavadam says
vipro ’dhītyāpnuyāt prajñāṁ
vaiśyo nidhi-patitvaṁ ca
śūdraḥ śudhyeta pātakāt.
A brāhmaṇa who studies the Śrīmad-Bhāgavatam achieves firm intelligence in devotional service, a king who studies it gains sovereignty over the earth, a vaiśya acquires great treasure and a śūdra is freed from sinful reactions
trayī na śruti-gocarā
śreya evaṁ bhaved iha
iti bhāratam ākhyānaṁ
kṛpayā muninā kṛtam
Out of compassion, the great sage thought it wise that this would enable men to achieve the ultimate goal of life. Thus he compiled the great historical narration called the Mahābhārata for women, laborers and friends of the twice-born.
ya edaM shrunyaa-Nityam yashchapi parikirtayaet |
Naa-aShubham praptyunaat kinCHIT-somutreH cha manavaH || (2)
Nothing inauspicious (or obnoxious) will befall the one who devoutly listens to the recital of
this sahasranama stotra daily and (the more so) the one who recites it wherever it may be,
whether in this world or the worlds beyond.
veedantago brahmana-asyat Kshatriyo vijayee bhavaet |
vaishyo dhana-SamRidhasya-chhuudra-sukhamvaapnuyaat || (3)
A brahmin he will attain the knowledge of the vedanta;
if a kshatriya, he will become a victorious warrior; if a vaisya, he will acquire immense
wealth, and if a sudra, he will remain happy
(Vishnu sahasranama phala Shruti)
For the learned and the unlettered alike in our country the Ramayana and the Mahabharata have for centuries been like their two eyes, pointing to
them the path of dharma
(Hindu Dharma The Universal Way of Life)
In summary, It seems that ,there is no inconsistency across scriptures bestowed by sage Vyasa (on this topic) .It only means that , some are entitled to access the same through shravana(told by the subject experts.)and not by adhyayana(study)