Trust me: you do want brifters; all of my vintage bikes have been retrofitted with these for convenience and safety. I suffered with downtube shifters for 25 years - never again.
As correctly identified by KCT1986, Shimano 7-speed shifters will index a 6-speed cogset with a 7400 rear derailleur. I suggest a Microshift 7-speed unit, which can be had for less than $100 on AliExpress or Fleabay. Of course, you'll need cable housing terminators, to cover the downtube shifter bosses. And please strip off all of your old cables and housing and replace with new. Plus a new 6/7/8 speed chain.
'Vintage' brifters, such as RSX? Good luck with these... I volunteer at a big-city Co-op, and we see many sets of these pass through our doors. Unless they were packed in a time capsule, there is only a snowballs' chance that a set of 25-year old Shimano brifters are functional or recoverable.