Nitpick: 1/10⁷, not 1/10⁶. They picked the power of ten that gave a reasonably-sized unit of length.
They also made things complex by then picking a unit of mass that’s inconsistent with that: a gram isn’t the mass of 1m³ of water, but of 1/10⁶ m³ of water (a cubic meter is 10³ liters, and a liter of water weighs 10³ grams)
The gram is the easiest to save by working backwards from Avogadro's number.
Avogadro's number is close to 24!, so we could redefine our mass unit as exactly 24! hydrogen atoms (or 4!!) and that comes out to within about 3% of a gram while being significantly easier to communicate to an extraterrestrial civilization.
Picking 1/10^9 would also have had the advantage of making the distance from the pole to the equator ~1 gigacentimeter.
Exploring a bit, picking 4 times that, as an estimate of the circumference, would give us a basic length unit of 4 cm (a pretty reasonable unit—bit over an inch and a half, fits right in with the variety of Chinese cun standards, for example). Then our immediate volume unit is 64 mL, which seems kind of small (on the scale of 2 floces), but ten of them make a decent "pint", so I think it can work—it's ~13% bigger than an imperial pint instead of ~12% smaller. the corresponding mass unit, at ~64 g, which again seems a bit small but manages to line up an average person's weight right around 1 kilo.
They also made things complex by then picking a unit of mass that’s inconsistent with that: a gram isn’t the mass of 1m³ of water, but of 1/10⁶ m³ of water (a cubic meter is 10³ liters, and a liter of water weighs 10³ grams)
Centimeter-gram-second (https://en.wikipedia.org/wiki/Centimetre–gram–second_system_...) really is superior in that sense (but of course, that’s relative to the arbitrary choice of using water to convert between mass and volume, and from that, length)