The circumstances of the revelation
Ibn Kathir doubted the narration you've quoted in his tafsir saying (My own translation take it carefully):
قال علي بن أبي طلحة ، عن ابن عباس : نزلت هذه الآيات في النجاشي وأصحابه ، الذين حين تلا عليهم جعفر بن أبي طالب بالحبشة القرآن بكوا حتى أخضلوا لحاهم . وهذا القول فيه نظر ; لأن هذه الآية مدنية ، وقصة جعفر مع النجاشي قبل الهجرة .
'Ali ibn Abi Talha narrated from ibn 'Abbas that these verses were revealed about the Negus and his companions, whom when Ja'afar ibn abi Talib -when in Etiopia- recited for them from the Qur'an , the tears rolled down their cheeks, but thgis statement is doubtful, as the verse was revealed in Medina, and the story of Ja'afar happened before the Hijra.
He also quoted other narrations:
Sa'id bin Jubayr, As-Suddi and others said that these Ayat were revealed concerning a delegation that An-Najashi (King of Ethiopia) sent to the Prophet in order to hear his words and observe his qualities. When the delegation met with the Prophet and he recited the Qur'an to them, they embraced Islam, cried and were humbled. Then they returned to An-Najashi and told him what happened.
He also commented on a strange narration from as-Sudi pretending Negus died on his way to Medina (as he apparently wanted to meet after this incident), but it is known that Negus died in Etiopia.
'Ata' bin Abi Rabah commented, "They were Ethiopians who embraced Islam when the Muslims who migrated to Ethiopia resided among them."
Qatadah said, "They were some followers of the religion of 'Isa, son of Maryam, who when they saw Muslims and heard the Qur'an, they became Muslims without hesitation."
(source: qtafsir, same link as above)
Comments on the verse
Note that Allah () here neither called the Christians followers of the Messiah, while he called 'Isa (), Masyh (Messiah) in other verses, nor did he call the Jews as followers of Moses (or any other prophet or messenger who was sent to them). So basically this verse is not a compliment for both.
Note that in Arabic nasrani نصراني (Christian) is far away from masiyhi مسيحي (follower of the Mesiah) which is the preferred why an Arabic Christian would be addressed with.
Also note that Allah mentioned the Jews beside the polytheist, as both rejected and opposed the message of our prophet () in a similar way and obstinacy.
The verse also clearly addresses the prophet () so this description will certainly apply for his time and at least the early days and centuries of Islam. What indeed needs to be checked is to what extent it would apply to our days? If we read most classical tafsirs we would come to the conclusion that this is the case, for example az-Zamakhsahri in his al-Kashaaf says, commenting on what the verse says about Jew:
ولعمري إنهم لكذلك وأشد
(My own translation take it carefully)
And indeed they are as described and (even) worse
Also note Allah uses an emphasize expression "You will surely find"
So does this verse apply to our time?
Ibn 'Ashur in his tafsir at-Tahrir wa-Tanwir comes to the conclusion that the Christians addressed in this verse were the Arabic Christians:
وظاهر قوله الذين قالوا إنا نصارى أن هذا الخلق وصف للنصارى كلهم من حيث أنهم نصارى فيتعين أن يحمل الموصول على العموم العرفي ، وهم نصارى العرب ، فإن اتباعهم النصرانية على ضعفهم فيها ضم إلى مكارم أخلاقهم العربية مكارم أخلاق دينية ، كما كان عليه زهير ولبيد وورقة بن نوفل وأضرابهم .
I'll try to translate this as follows:
The apparent meaning of those who said "We are Christians", is that these attributes (manners) ate a description of all Christians from the perspective that they are Christians, so this apparent meaning should be taken according to custom, this means they are the Arabic Christians, as they followed Christianity, even if their fellowship was rather weak (or incomplete) beside the praiseworthy virtues of Arabs(*), religious virtues as it was the case for Zuhair, Labid, Waraqa ibn Nawfal and people of their kind.
(*) such as hospitality, keeping a covenant, sense of honour and repudiation of injustice, firm will and determination, forbearance, perseverance and mildness ... (For details read in ar-Raheeq al-Makhtum/Chapter:Aspect of the Pre-Islamic Arabian Society/Ethics)
In ar-Raheeq al-Makhtum of Safiu ar-rahman al-Mubarakpuri you may find the following description of pre-Islamic Arab Christians:
Christianity likewise opened its doors wide to polytheism, and got too difficult to comprehend as a heavenly religion. As a religious practice, it developed a sort of peculiar medley of man and God. It exercised no bearing whatsoever on the souls of the Arabs who professed it simply because it was alien to their style of life and did not have the least relationship with their practical life (Chapter: Religions of Arabs)
Ibn 'Ashur says that this goes ahead with a tradition narrated by at-Tabri, al-Wahidi and many tafsir authors quoting that 8 (Arabic) Christians from a-Shaam were in Etipopia and joined a group of 62 monks who visited Medina with the Muslims who where in Ethiopia and there they listened to the Qur'an and accepted Islam (This was in the year 7 a.H.). Names of these monks and Arabic Christian were also quoted and that some of them were able to speak Arabic. So their story was told in verse (5:83).
Ibn 'Ashur finally concludes saying:
والمقصود أن الأمة التي فيها أمثال هؤلاء تكون قريبة من مودة المسلمين .
My translation take it carefully:
And the meaning here is to say that the nation which has people as described (in 5:82-83) will be close in their affection to Muslims.
From this we may conclude that the verses may apply to all nations who either seem to be described by the description of Jews and polytheists in their hatred towards Muslims or those who are described by the attributes of Christians in their affections towards Muslims generally.