Advaita Vedānta was the philosophy taught by Ādi Śaṅkarācārya, and it is still taught today by his successors. One of them, Sri Bharati Tirtha Mahaswamiji of Sringeri Sharada Peetham teaches that nāma japa is suitable and beneficial in Kaliyuga. If Advaita and the practice of nāma japa are perfectly compatible to a Jagadguru/Śaṅkarācārya, could they also be compatible to somebody who is not yet Self-realized?
Logic is necessary and helpful in mundane matters whereas direct experience is important in spiritual matters. Perhaps logic should be reserved for mundane matters apart from spiritual matters?
I suppose you could manually let go of negative thoughts.
When it comes to a word producing the same benefits as nāma japa, I don’t know if it could happen. What I do know is that the Śiva Purāṇa, for instance, states that Śiva nāma japa reduces mountainous heaps of pāpam to ashes, takes away worries, removes the fear of Yamarāja and Naraka, and is the raft that takes one across the ocean of worldly existence to liberation. The Padma Purāṇa says similar things about Rāma, the highest name of Nārāyaṇa. I haven’t yet found a scripture or Self-realized person who says such things about words such as maitrī, śānta, or karuṇā.
There is always the practice of aṣṭāṅga yoga. Dhāraṇā or ‘one-pointedness’ is an aspect of it that follows pratyāhāra or ‘gathering toward.’
If that is your experience from chanting ‘Jai Hanumān,’ then that particular japa is not the best one for you if you don’t want to feel disconnected from your family.
The brahmacarya of Hanumān is not the brahmacarya of a husband. Dharmaśāstras and other texts teach that a married man is to approach only his own wife. That is the brahmacarya of a husband. Which would Hanumān approve of in you?