No, not 'all' : Lance used one on some of the climbing days on some of his earlier Tour wins.
On flatter days in the same race, he used a lefthand STI brifter
Whether it actually made much difference is moot - may just have been publicity or out-psyche-the-opposition stuff.
In later Tours, he used STI's both sides even on climbing bikes, as Trek had by now made a bike under the minimum weight limit, needing to be ballasted-up rather than weight saved, and STI front-shift and trim technology had improved by now.
Personally, if I had a retro bike which I dug-out on the occasional sunny Sunday, rather as people drive around in classic 1950's sportscars, then obviously it would have the correct period downtube shifters.
But for a modern bike for everyday use, no chance !
I can remember hitting a pothole years ago, with only one-hand on the bars and one down on the shifter, and enduring a huge tank-slapper before I got back under control, my ears filled with the sound of a large diesel engine growling along just behind me...