IMO

'road' is (as you say) 'connecting link' between 2 places (towns, cities, villages, A-B, home-school,...).

'street' is a different concept expressing 'a passage/thoroughfare' through an area (eg. business or shopping district) of activities. 'Avenue' is a broader concept of thoroughfare through many more types of areas eg. park, beach, commercial enterprise (as in 'way to make money'),...

The observation that roads are long and streets (and avenues) are much shorter is correct but the length is not a criterion for differentiating roads from streets. For us, roads or streets are 'what they are named' (or known).