Yes , Shree Krishnas argument is correct. There are multiple interpretations available of this verse and his saying to Arjuna. What is actually its inner meaning and what is Shree Krishnas actual message to us.
हतो वा प्राप्स्यसि स्वर्गं जित्वा वा भोक्ष्यसे महीम्।
तस्मादुत्तिष्ठ कौन्तेय युद्धाय कृतनिश्चयः ।।BG 2.37 BG
hato vā prāpsyasi svargaṁ jitvā vā bhokṣyase mahīm
kaunteya yuddhāya kṛta-niścayaḥ
O son of Kuntī, either you will be killed on the battleﬁeld and attain
the heavenly planets, or you will conquer and enjoy the earthly
kingdom. Therefore, get up with determination and ﬁght.
English Translation of Ramanuja's Sanskrit Commentary By Swami Adidevananda.
2.37 If you are slain in a righteous war by enemies, you shall thery attain supreme bliss. Or, slaying the enemies, you shall enjoy this
kingdom without obstacles. As the duty called war, when done without
attachment to the fruits, becomes the means for winning supreme bliss,
you will attain that supreme bliss. Therefore, arise, assured that
engagement in war (here the duty) is the means for attaining release,
which is known as man's supreme goal.
See the real meaning of this shloka beautifully explained by Shree Ramanuja. He is explaning us that this verse is intended towards both Arjuna as well as the people who are following path of Karma Yoga. Here Krishna is giving stress on Karma i.e. our duty , Swarga is symbolic of supreme bliss which one will attain by following his duties without attaching to its fruits. And Shree krishna here is perfectly right , because Niskam Karma Yoga is is the means for attaining release.
Answer To Your Argument No .1
- Since Krishna incited the Pandavas to commit war-crimes ?
No he is not encouraging Pandavas to commit war-crimes , he is just asking them to wage a war against Adharma , and win a righteous war for establishing of Dharma. To set an example. As it is duty of kshatriya Varna to protect Dharma even through war. So No war crimes here.
Krishna Himself. Then He said to Arjuna, “They are already killed. You just become My instrument.” So no war crimes again here , because Bhagvanta already killed them. Apart form that Krishna also told Arjuna that death of the body is not the death of the soul ,where as our pure self or Atma is non perishable.
So Pandavas were just the cause
तस्मात्त्वमुत्तिष्ठ यशो लभस्व जित्वा शत्रून् भुङ्क्ष्व राज्यं
समृद्धम्। मयैवैते निहताः पूर्वमेव निमित्तमात्रं भव सव्यसाचिन्।। BG
tasmāt tvam uttiṣṭha yaśo labhasva jitvā śatrūn bhuṅkṣva rājyaṁ
samṛddham mayaivaite nihatāḥ pūrvam evanimitta-mātraṁ bhava
Therefore get up. Prepare to ﬁght and win glory. Conquer your enemies
and enjoy a ﬂourishing kingdom. They are already put to death by My
arrangement, and you, O Savyasācī, can be but an instrument in the
Answer To Your Argument No .2 - These are not the only two alternatives - Arjuna and/or his kinsmen might have been horribly wounded, but alive and lived a long time in terrible pain.
Yes Pandavas were got wounded in the war and they suffered the pain due to great loss of kinsmen and people at the end.
Answer To Your Argument No .3 -Assuming weapons like Brahmastra weer real - there might have been an ecological catastrophe from the war.
Scriptures tells us that there were mass destruction , but see what Gita is telling us about disasters or destruction.
अव्यक्तादीनि भूतानि व्यक्तमध्यानि भारत।
अव्यक्तनिधनान्येव तत्र का
परिदेवना।। BG 2.28।।
avyaktādīni bhūtāni vyakta-madhyāni bhārata
tatra kā paridevanā
All created beings are unmanifest in their beginning, manifest in
their interim state, and unmanifest again when annihilated. So what
need is there for lamentation?
English Translation Of Sri Shankaracharya's Sanskrit Commentary By Swami Gambirananda.
i.e. with regard to beings which are objects of delusion, which are
invisible, (become) visible, (and then) get destroyed!
So destruction and again manifestation both are part of great cycle. This destruction was planned by Bhagvanta himself for the re-establishment of dharma.
And there is presently no mentioning of any ecological disaster happened during or after the war apart from lots of wood piles were used to burn the bodies of the dead. But Jungles and trees can grow again. And the animals which were killed in the battle attained swarga.
Conclusion - So krishna'
s argument is correct ,and its been well verified and commented by lots of Scholars and Acharyas. There is no doubt about correctness of this saying by Krishna.