Adhikāra presupposes 1. śakti (or sāmarthya), as discussed in previous posts. What else does it presuppose? Adhikāra for sacrifices presupposes a condition (nimitta) and/or the 2. desire for a goal. Both are necessary conditions, as it becomes obvious when animals are said to potentially have the adhikāra to sacrifice, since they are desirous (arthin), but to be excluded from it because of their lacking the relevant ability. The same discussion is found in the commentaries on UMS 1.3.34–38 in relation to śūdras’ adhikāra (e.g., Meghanadāri thereon: arthitvasāmarthyādisadbhāvāt [śūdreṣu] adhikāro ‘stu).
What else? Notice the ādi in Meghanadāri, which is repeated also elsewhere in his commentary (e.g.: atra saṃśayaḥ— kiṃ śūdrasya brahmavidyāsv adhikāro ‘sty uta neti. tadarthaṃ kiṃ tasyārthitvādīni santy uta neti). Meghanadāri lists also the 3. adhikāra for prerequisites, such as the fact of having set the fires (that is, having performed the initial ceremony that starts one’s life as a performer of sacrifices). Śūdras lack the adhikāra for sacrifices, he says, because they lack the adhikāra for the actions sacrificing presuppose (brahmavidyā and agni-setting) (vaidikakarmasu śūdrasya vidyāgnirahitatvenānadhikāraḥ samarthitaḥ).
Thus, adhikāra requires: 1. ability, 2. desire, 3. adhikāra for prerequisites.
Another interesting point is that adhikāra applies to a certain way to achieve a goal, not to the goal itself. You are disabled and cannot perform sacrifices to go to heaven? You can go to heaven through recitation of the Veda. You are a śūdra and cannot read the Upaniṣads? You can still learn about the paramātman (!) through itihāsas and purāṇas. An opponent says that veneration (upāsanā) still needs its auxiliaries (aṅga) and that these are sacrifices, to which the śūdras lack the adhikāra. Hence, due to the rule discussed above, they also cannot have the adhikāra for upāsanā. Meghanadāri replies that this is not the case, because, like in the case of fixed sacrifices, one does not need to perform all auxiliaries.
(Many thanks to Manasicha Akepiyapornchai for sharing the images of Meghanadāri’s commentary)