Any manuṣya, regardless of liṅga or jāti (including cāṇḍāla), has the potential to attain svargagati or narakagati (upon mṛtyu), depending on that person's karman.
karama pradhāna bisva kari rākhā /
jo jasa karai so tasa phalu cākhā //
~ Rāmacaritamānasa (2.217.2)
The śloka of OP's concern is Uttarakāṇḍa (76.8) -
gṛhāṇa ca varaṁ saumya yaṁ tvamicchasyaridam /
svargabhāṅ nahi śūdro'yaṁ tvakṛte raghunandana //
Śambūka's tapa was geared towards going saśarīra (with body i.e. sthūlaśarīra) to svargaloka & attaining devatva. This is clearly mentioned in Uttarakāṇḍa (76.2), where Śambūka says -
śūdrayonyāṁ prajāto'smi tapa ugraṁ samāsthitaḥ /
devatvaṁ prārthaye rāma saśarīra mahāyaśaḥ //
Śambūka's saṅkalpa regarding his tapa, is mentioned in Padma Purāṇa (1.35.83-84) too -
yadetadbhavatāproktaṁyonaukasyāṁtutetapaḥ /
śūdrayoniprasūtohantapa ugraṁ samāsthitaḥ //
devatvaṁ prārthayerāmasvaśarīreṇasuvrata /
namithyāhaṁvadebhūpadevalokajigīṣayā //
Based on these sources, we can say that Śambūka's saṅkalpa seems similar to that of Rāma's ancestor Triśaṅku (Bālakāṇḍa 57.11). However, Triśaṅku wanted to accomplish that via a yajña (gaccheyaṁ svaśarīreṇa devatānāṁ parāṁ gatim)and didn't cause the death of innocent(s), by anuṣṭhāna of any niṣiddha tapasyā, unlike Śambūka, whose tapa resulted in the death of innocent [Uttarakāṇḍa (74.28-29) ; Padma Purāṇa (1.35.35-37)].
In Rājyakāṇḍa (1.10.95-100) of Ānandarāmāyaṇa, it's mentioned that Śambūka's tapa didn't result in the akālamṛtyu of just one Brāhmaṇaputra, but of 6 other people, all 7 were revived upon Śambūka receiving mṛtyudaṇḍa from Rāma (Ibid., 1.10.119-120). Their deaths were revealed after announcement of Brāhmaṇaputra's death.
Uttarakāṇḍa (76.8) is referring to the destruction of the possibility of actualization of Śambūka's saṅkalpa, because upon being slain, he had lost all chances to be able to attain saśarīra svargagati, an act which is essentially against the vidhāna of prakṛti. But upon being being daṇḍita by Rājā Rāma himself (who is dharmavigraha), his pāpa was essentially destroyed, and so he attained svarga upon mṛtyu.Here, the mṛtyudaṇḍa bestowed by Rāma serves a treble purpose i.e.- (i) of destroying the pāpa (so that Śambūka requires no prāyaścitta, etc. or facing the karmaphala of his pāpa at a later stage in this life or next) ;
(ii) of bestowing the daṇḍa on Śambūka for the pāpa he committed ;
(iii) of reviving the ones who faced akālamṛtyu due to Śambūka's pāpa.
In Rājyakāṇḍa (1.10.115) of Ānandarāmāyaṇa, Rāma tells Śambūka that he would attain vaikuṇṭhagati after receiving mṛtyudaṇḍa bestowed by him (Rāma).
tavāpīyaṁ kathākīrti smariṣyanti madaṅghrijāḥ /
tvaṁ mayā nihatastvadya vaikuṇṭhaṁ prati yāsyasi //
In Raghuvaṁśa (15.53), Kālidāsa also states that Śambūka attained sadgati, upon being daṇḍita by the Rājā, and not because of his kaṭhoratapasyā, which was itself against dharma. -
kṛtadaṇḍaḥ svayaṃ rājñā lebhe śūdraḥ satāṁ gatim /
tapasā duścareṇāpi na svamārgavilan̄ghinā //
Mallinātha, in his ṭīkā Sañjivinī on Raghuvaṁśa (15.53), cites Manusmṛti (8.318), which states -
rājabhiḥ kṛtadaṇḍāstu kṛtvā pāpāni mānavāḥ /
nirmalāḥ svargamāyānti santaḥ sukṛtino yathā //
Therefore, the incorrect claim that Śūdras are unable to attain svargagati, is shown invalid.
