Your right. By no means cut down to the dead wood. The wood parts cannot put out new growth.
Here's what I would try if it were mine. Anyone else is welcome to reflect on it and argue otherwise. I'm by no means sure of success, but here's my reasoning.
It missed its annual pruning, and that's almost certainly why is has dead areas and is so splayed out and not at all attractive in a natural lavender way. I don't want to just leave it like that. That's the road to death.
But French lavender is more sensitive to pruning error than English. I would tentatively:
Cut away all the mostly dead splayed out branches, even if they have healthy green ends. I mean those on yours that are hanging over the rim of the pot. You don't want those shaping your plant.
It's hard for me to see clearly, but it looks like the rest of yours has branches brown on their lower halves, with plenty of green above. And it looks to me like you have four to six inches of green stemmed recent growth above the wood. If that's right, I think I'd prune so that there was only about three inches of healthy, green (not woody) stem.
That will likely stimulate some new growth and start getting the plant back on schedule. Frost is probably not an issue for you, so I'd just watch them over the winter. If I guessed right, the new growth will pick up, it will flower in spring, and you can then go to a standard pruning schedule where, after the bloom, you shave it back down to the two or three inches of green again. Or if there's not enough growth to make that feel okay, deadhead the flowers. I'd probably cut off flowers the next year, anyway, to conserve energy.
If it was a big old neglected woody lavender, I'd definitely use a rotation, hard pruning it by thirds or quarters, because a hard pruning of the whole thing could kill it. I'm frankly unsure about yours, so I think I'd just be careful to leave plenty of green. The woody parts will not put out new growth, not do they breathe. In the wild, lavender has good roots, so it can send up shoots if something very bad happens to it. I wouldn't gamble on that happening with a younger plant.
Now, it's not going to be a pretty plant for a long while. I might consider, after it has time to get going, repotting it, burying the bare woody stems. That could be late winter or early spring, if you think it's looking healthier. If that goes alright, you can begin shaping it in the summer.
I have a French lavender I just received and planted in a wide pot. I don't know its history, but it looks like it has recently put on new growth, so I will probably give it a modest pruning in August to prevent what you have now.
And do not fertilize. French lavender evolved for low-fertility soils and will get leggy and weird with too much nutrient.
