Public doesn't care: there's no draft and the deaths, if there are any, will be in the tens, not the tens of thousands.
Tell that to the survivors of the Lang Vie Special Forces Camp in Quan Tri province, or the survivers of 2/7 Cav at LZ Albany during the Ia Drang battles made famous by Hal Moore.By the time this TL diverges there was no money available for military operations in Vietnam and there were no significant forces in Vietnam outside of the US controlled area in Saigon near the Embassy.
The US Armed forces never actually lost a battle with the NVA or the VC. The Vietnam War was lost at home. And by 1974/75 with 58,000 American soldiers dead and nothing to show for it the war was HUGELY unpopular. Any move made by Goldwater to get the USA reinvolved in Indochina on the ground would have been political suicide and possibly resulted in his impeachment. It most certainly would have resulted in civil unrest at home.