Why does/would a "Bodhisattva" burdens himself and instigate others as well?

FYI I tried to follow on this comment which recommended explaining "the 4 types of nirvana."

There's a description of that in this video, Four types of nirvana starting at about time 13:05.

• Natural nirvana -- the emptiness of the mind
• Nirvana with remainder -- an arhat (the "remainder" being their body)
• Nirvana without remainder -- an arhat after death

Some schools believe that an arhat's mind and body cease at death, which begs the question "who has nirvana?" The venerable's (Gelug) school teaches that, instead, when an arhat dies they shed their polluted body, acquire a mental body, and reside in a Pure Land in meditative equipoise (on emptiness) until the Buddha arouses them and says they need to enter the Boddhisatva path: at which they have to start on the Boddhisatva path (even though they've realized emptiness before) and acquire the great merit of a Boddhisatva. And that's why (she says) we're encouraged to enter the Boddhisatva path directly without become arhats first ... because that (entering the Boddhisatva path directly) is the quicker (than becoming an arhat first and being in meditative equipose for a while) way to enter full awakening, accumulate all the merit that you need to accumulate to become a Buddha.

There are many motivations.

One of the main motivations is loving kindness. You do this to the benefit of the others.

Though this might not be generally considered, a small strain of sense of challenge, achievement, accomplishment or to achieving the ultimate can also be present. These may fade away depending on the stage you are in but surely a motivation early on to wish for Buddhahood. Buddhahood is at the pinnacle of working towards a most perfect form of existence. This on it's own right becoming.

Also external influences can motivate this. I have heard when there were no Buddhas for a long time some beings who are is some stage of sainthood in higher planes influence possible earthlings to aspire to become a Buddha by telepathic influence and planting the seed or initial thought of aspiration.

A Bodhisattva 'instigates' others because he wants what is best for others (full enlightenment together with its causes). From this viewpoint, 'abiding nirvana' is thus not what is best for others, because it is not the one final result of the one final vehicle (i.e. the Mahayana path).

There are two obscurations:

1. Afflictive obscurations - all the afflictions - which prevent one from achieving abiding nirvana, and
2. Knowledge obscurations - the imprints left by previous moments of ignorance - which prevent one from achieving perfect enlightenment, omniscience.

If he completely abandoned afflictive obscurations (which an arya bodhisattva does anyway from the 8th bhumi), the practitioner would be free from the conception of inherent existence, but he would not be free from the appearance of inherent existence because such a mistaken appearance is caused by knowledge obscurations. A bodhisattva does not aim at merely abiding in 'abiding nirvana', but wants to be completely free from all mistakes and thus be able to enact the welfare of others.

The bodhisattva does not set out to merely get rid of afflictive obscurations and achieve abiding nirvana. He set out, from the entry into the Mahayana path, to get rid of knowledge obscurations and achieve what Mahayana call 'full buddhahood', 'perfect enlightenment', 'omniscience', 'the four kayas', 'non-abiding nirvana'.

The wisdom directly realizing emptiness is said to be 'the mother of all three types of enlightenment - that of the shravaka, the pratyekabuddha, and the bodhisattva' because it is a cause of enlightenment. However, it is a Mahayana belief that full enlightenment (that of the arhat-bodhisattva, an arya buddha) can not be achieved by cultivating the wisdom-side alone. One needs the method-side, namely bodhicitta, the six perfections, and so forth.

Knowledge obscurations can only be removed by way of developing the two wings that are method and wisdom. When the wisdom directly realizing emptiness is conjoined with bodhicitta, that wisdom becomes what is called 'ultimate bodhicitta' and that puts an end to what prevents one from achieving omniscience, the omniscient mind of a buddha, buddhahood, perfect enlightenment. That is why a bodhisattva cultivates bohdicitta, which is 'the wish to achieve enlightenment for the benefits of all sentient beings'.

The classical analogy goes: the wisdom realizing emptiness is like an axe while bodhicitta is like the strong arm. You need both the axe and a strong arm to cut the tree (that is here knowledge obscurations).

We say that from the path of seeing, an [arya] bodhisattva comes back in samsara but is not wandering in samsara. So his aggregates are not contaminated anymore, and there is no 'true suffering' involved. It is not the case that he will suffer for eternity. This is because he does not come back under the power of karma and afflictions, but under the power of compassion and aspirational prayers.

Why does/would a "Bodhisattva" burdens himself with the choice of becoming a Buddha (avoiding extinguishing dukkha) and instigate others as well to do so?

Many people misunderstand Nirvana as permanent happiness or permanent absence of any trouble - which is impossible due to Impermanence. So Bodhisattva explains that such "Nirvana" is impossible, and invites all who has ears to hear to partake of true Nirvana, which is suchness - true extinguishing of dukkha by not clinging to what passes and not rejecting what is and not craving for what is not.

Second, if one wants to teach others Dharma, dukkha will inevitably be generated due to mismatch between "what is" (the confused student) and "what should be" (the student that understood). So in order to teach one has to be Okay with dukkha - which in itself is extinguishing dukkha.

So Bodhisattva really only consistently applies Dharma to its completion.

Given that this world is full of "Bodhisattvas", there is not one being who is not just here to help others out,

This statement is false. There are millions if not billions of selfish people here - and you know this. Sarcasm is a form of false speech too.

Or what is the reason to instigate others to do such a work?

Compassion.

Or do they assume that there are no teachings which lead them out for now?

The desire to get "out" is the desire that Bodhisattva overcomes to attain the non-attainment of suchness, the true Nirvana.