its funny, but I always shifted only with my right hand for both dt levers, often both at the same time with diff fingers, but it would still seem to me to intuitivley want the front der to be a left side bar end, to separate the duties and free up the right hand for rear only sihfts.
When I used a Suntour high normal front derailleur where pushing both down tube levers forward put you into the highest gear, I usually used the right hand for both front and rear shifting. Cresting a hill top I would often reach down and push both levers forward simultaneously to go into my highest gear. But when my front derailleur was a conventional derailleur (pull the down tube shifter back for the highest gear), then I would usually use my left hand for the front and right hand for the rear.
And then there is my rando bike, a Campy brifter for the rear and as I mentioned above a downtube friction shifter for the front. The downtube shifter was intended to be temporary while I decided what to use permanently, but I built up that bike in 2016 and temporary is slowly becoming permanent. See photos. I usually use the left hand to shift the front derailleur. The downtube shifter is a vintage Huret that I picked up for $5 USD.