You're forgetting about STAB. Psychic still does more damage to dragonite than ice beam anyways. Because Slowbro is a psychic type he always gets the +25% damage but even without it psychic still wins because ice beam is low damage. Also having +25% vs all enemies (including dragonite) is better than +25% against just dragonite.
GamePress
Raise the status of Slowbro with water gun+ice beam
Currently the best offense move set for slowbro is water gun+physic
The problem is, we have vaporeon leading water type, and exeggutor leading physic type. This move set makes slowbro always a standby fighter only.
Due to the popularity of dragonite, ice is highly needed. Other than blizzard(only lapras, dewgong, jynx, cloyster have and got STAB), the second best ice move 'ice-beam', owned by slowbro, blastoise, golduck, and some other weaker species, is also a very good option against dragonite.
Lapras is always the elite option but I guess it is extremely hard to get one with frost and blizzard, and some more, high IV. Comparing the rest ice pokemons, dewgong has relatively low base attack(150+), cloyster has relatively low base stamina(100/110?). for Jynx I don't need to say anything.
But comparing the rest three non-ice type species abovementioned, slowbro stands out with overall best stats and highest possibility to catch one with high iv.
Though water gun is not effective to dragon, but it charges damn fast to cast ice beam, slowbro, the most tanky, allows more beam to be casted. You will really make dragon opponent feel the pain.
I suggest the ranking of dragonite counter is lapras>slowbro=dewgong>blastoise>golduck=cloyster>jynx
