I've turned the bottle upside down and filled it with water. The water runs easily through the filter without squeezing. Gravity is sufficient.
I think I could probably scoop water from a lake or stream with the bottle attached to the filter. Would probably get some dirty water on the filter, however. Not sure if it would matter. I could wipe the filter dry and then discard the first few ounces that came through the filter.
I alreay use the bottle (or one like it) as a stove carrier and have done so for many years. Carrying the stove assembled reduces the risk of cross threading it in the field during assembly.
I think your cord idea would work nicely too.