First time I saw this approach was in the Tal-Botvinnik return match in 1961, my thought was "it wastes a move heading for the French Defence," but it has some sting in it.
Black isn't really giving away a pawn, White can't hold the extra pawn without development issues (...Nd7 pretty much ensures the pawn comes back, if not immediately, then soon, if it worries you, though ...Nc6 will exert more pressure on White's split pawns).
What you're opting for, when keeping this line closed, over Bf5 is keeping your light bishop (in the Bf5 lines, White can pretty much force that off the board at will) even though it gets blocked in with the e6-pawn, moving some of your strategic ideas towards the French and away from the C-K.
You're giving up a tempo, yes, but the game becomes sharper. White becomes the first person to release the tension in the center, which frees up Black's game a little, so you're looking to capitalize on the freedom. Like any good French player, you need to be prepared to keep that light-square bishop at home for a long time while you play against the e-pawn.
Meanwhile, if the idea of letting the pawn hang for a while doesn't bother you, ...Nc6 instead of ...e6 can lead into some Queen's Gambit like positions with the colors reversed (keep aware of the tactical shot ...Qh4, you might be able to use it to threaten stragglers on the Q-side, like Keith Arkell did in a recent Senior Championship. Another of Arkell's ideas is to quickly swap the light Bishop for White's Knight before pushing ...e6, if the opportunity permits, at which point Black has a better pawn structure than White, and may even have the better game going forward; it's the French Defence without the bad French Bishop. In the ...Bf5 lines it's usually a Bishop trade, oftentimes trading a bad Bishop for a useful Knight is the better option.
It's really not all that bad (I find it hard to conceive that anything Botvinnik came up with is truly bad) but it caters to a different style of play than the more fashionable ...Bf5 lines. Arkell, Anand, Ponomariov, and Tarjan have all essayed it recently. Like pretty much any opening, White gets a slight pull if they play it right, but you're always "playing uphill" as Black, so don't let it worry you.
That's a few ideas for you to look at; go have a look at games from the French Advance and see what ideas you find there that appeal to you. Keith Arkell seems to be a common practitioner of the line today, have a look at some of his games from the Pro Chess League. There's scope for play here, if it's to your taste. But only you can answer for that.