Agreed. Haste is giving you an Attack action on your turn. Ordinarily, that should trigger the "Moreover" text of BS. However, Haste is constraining the use of the action to "one weapon attack only".
A principle that might be appealed to here is that words in rules should be given meaning, if possible. If Haste is not really granting an Attack action, but instead granting a weapon attack only, then why does Haste contain words that expressly grant an Attack action?
I think one must conclude that Haste is really, truly granting its beneficiary an Attack action. Haste constrains the action, but the constraint can be taken to be silent on other words in the rules that revise what can be done with your Attack action. For example, if some feature had words that said - "When you use an Attack action on your turn to make a weapon attack, then..."
Would you agree that if one allows the cantrip, then to be consistent one must also allow Battlemasters to sub in Commanders Strike and Monks to throw in a Martial Arts unarmed strike?