Salamun Alaikum brother,
The Quran is simply informing believers to send blessings on the Prophet and to greet him with suitable greetings. God and the angels even send blessings on the believers (33:43). So there is no outright exclusivity for blessings or greetings here for the Prophet.
It is like me saying to you 'God bless you dear ..." and when I meet you I would say 'Peace'. God clearly instructs believers to greet each other with suitable greetings of peace and to return them (4:86) which is no different for the Prophet.
The reason for specific mention is so that Prophet Muhammad's 'special capacity' as a messenger of God and spiritual guide is clearly recognised and established amongst his people.
As the Prophet would have clearly had people older than him, old frank friends, other peers etc , God is clearly demarcating boundaries and making it absolutely clear that Prophet Muhammad's capacity was far more deserving of respect than how people treated each other. Friends, colleagues and elders can often take liberties in speech and dealings.
The boundaries are clearly demarcated by the Quran.
049:002
"O you who believe! Raise not your voices above the voice of the Prophet, nor speak aloud to him in talk, as you may speak aloud to one another, less your deeds become vain and you perceive not"
024:063 (part)
"Make not the calling of the messenger among you as your calling one of another..."
Obviously when the Prophet was alive, it would have been possible to greet him with suitable greetings or not to speak aloud to him. Now all we can do is send suitable blessings on him as he is not alive today.
The question intimated by your friends as to how one sends blessings without Ahadith is wholly inappropriate in my personal opinion and has absolutely no warrant. Does one really need to be taught how to bless someone sincerely with heart and soul? To ask the Quran to ratify a 'particular' belief, saying or practice is nonsensical. It is the role of 'practice' or 'belief' to find support from the Quran.
It is akin to saying, God says 'Eat and Drink' and then to seek justification for a specific kind of drink.
I hope that helps, God willing.
Joseph.