Firstly: The verse says:
Allah does not forbid you ... from being kind towards them
In simpler words, this means:
Allah permits you ... to be kind towards them
Permission is not the same as obligation, rather it implies a choice. If it is merely permissible to be kind towards them then it is also permissible to be harsh towards them and to fight them. And it is permissible to be harsh to some of them and kind to some of them based on what suits the interest of the Muslims.
Hence this verse does not forbid offensive jihad, at most it would make it optional.
Secondly: The verse says:
those who do not fight you
You are assuming that it means those who do not 'initiate' fighting. However this understanding is contrary to the other verses in the Quran and with the Sunnah of the Prophet and the Caliphs.
Rather it means those who do not fight when we declare jihad. They are:
The Ahl al-Dhimma who when given the ultimatum in Jihad, chose to pay Jizya rather than to fight.
The Mu‘aahideen who have an extant treaty which binds them to abstain from fighting us.
The women and children, and possibly the monks and the disabled etc. who do not fight as they can not.
Some of the exegetes have also said that it means the Muslims who live in the Dar al-Harb and have not migrated.
The evidence of this is the hadith about the of reason for revelation of this verse:
عن أسماء ابنة أبي بكر ـ رضى الله عنهما ـ قالت قدمت على أمي وهى مشركة في عهد قريش، إذ عاهدوا رسول الله صلى الله عليه وسلم ومدتهم، مع أبيها، فاستفتت رسول الله صلى الله عليه وسلم فقالت يا رسول الله، إن أمي قدمت على، وهى راغبة، أفأصلها قال " نعم، صليها ".
فَأَنْزَلَ اللَّهُ تَعَالَى فِيهَا: {لاَ يَنْهَاكُمُ اللَّهُ عَنِ الَّذِينَ لَمْ يُقَاتِلُوكُمْ فِي الدِّينِ}
Asma 'bint Abi Bakr said:
During the period of the peace treaty of Quraish with Allah's Messenger (ﷺ), my mother, accompanied by her father, came to visit me, and she was a pagan. I consulted Allah's Messenger (ﷺ), "O Allah's Messenger (ﷺ)! My mother has come to me and she desires to receive a reward from me, shall I keep good relation with her?" He said, "Yes, keep good relation with her."
...
Then Allah revealed: 'Allah forbids you not with regards to those who fought not against you because of religion, and drove you not out from your homes, that you should show them kindness and deal justly with them'
— Bukhari 3183 and 5978
Which indicates that this verse was revealed about a woman, and during the peace treaty of Hudaibiyah.
Thirdly: Some of the exegetes have claimed that this verse is abrogated. That is because this verse was revealed before the conquest of Makkah. Whereas the final laws of jihad were revealed after that in Surah at-Tawbah.
And the verses of Surah at-Tawbah endorse fighting and harshness against disbelievers in general without differentiating - such as:
يا أيها النبي جاهد الكفار والمنافقين واغلظ عليهم ومأواهم جهنم وبئس المصير
O Prophet, fight against the disbelievers and the hypocrites and be harsh upon them. And their refuge is Hell, and wretched is the destination.
— Quran 9:73
With the only exceptions in Surah at-Tawhab being those who have treaties see 9:29, 9:4, 9:6.
Ref: Tafsir al-Qurtubi
