The thing is, the density decreasing with height, which is caused by the reduced surrounding pressure, does mean the air is spread out more... but likewise also means the water vapor must take up a larger space for the same amount of molecules too. And so that density/pressure change basically doesn't mean any "more room" for water vapor (the expanding existing water vapor volume takes it all).
Looking deeper, we often talk about it all loosely in terms of "room". But truly temperature decrease isn't directly changing the room (yes there can be a density/volume change, but also a pressure change), but means the molecules are moving slower and so are more likely to condense together.
As you lift unsaturated air, the amount of molecules of water vapor and air stay the same.
The total and vapor densities decrease by equal ratios due to the reduced surrounding pressure, which also brings a temperature decrease because energy is used to expand to fill the larger volume. And both of those mean their internal pressures fall by the ideal gas law.
But as Poutnik indicated, the saturation vapor pressure (what is often poorly alluded to as the "room" available) is basically only a (direct) function of temperature in our atmosphere. So it decreases as temperature decreases, and isn't significantly affected by any isothermal processes. And while a lot of variables change in lifting air (the dew point and specific humidity both decrease due to the density drop/parcel expansion... and the vapor pressure drops due to the temperature decrease), the mixing ratio doesn't change, and so the only notable cause for change in relative humidity is the decreasing temperature's impact on saturation vapor pressure (the amount of "room" available until saturation, which isn't so much about room, but freedom).