Gallium is found everywhere, it doesn't make sense to actively search for it because it doesn't concentrate anywhere.
That's the whole problem. There is little that distinguishes bauxite ore from my backyard in terms of the amount of gallium that can be extracted.
The advantage of extracting it from certain ore waste streams is two-fold even though they don't contain much gallium. First, the chemical processing cost varies with the chemistry of the rocks you extract the gallium from, and certain types of ore waste such as bauxite, zinc, etc are cheaper to deal with. Second, these rocks have already been dug out of the ground as part of a mining operation, which is much cheaper than strip mining an arbitrary place to extract the same trace quantities of gallium -- you get to free-ride on the extraction costs of the primary mineral someone already paid for. If it doesn't matter where you dig, then all you can really optimize for is the processing cost of where someone already dug.
It would make no sense to increase bauxite production for the purpose of gallium production. Extracting gallium from bauxite is only economical to the extent that there is healthy demand for the aluminum produced from that bauxite. This is common in mining operations -- a secondary mineral that cannot be economically mined by itself becomes profitably extractable from the same ore if and only if the primary mineral is sufficiently profitable. Many less common metals are produced solely via secondary extraction because they cannot be profitably mined directly even when they concentrate.