Golf cart paths shown as walkways (in pink) and difficult to read.

Have you considered using access tags on a highway=service or highway=unclassified tag? Sounds like motor_vehicle=no+golf_cart=yes on one of those will do what you need. highway=path with golf_cart=yes` may be another more appropriate tag. The tag is usually applied to unpaved routes, but it's meaning ("A route open to the public which is not intended for motor vehicles, unless so tagged separately.") applies fine here and is devoid of specifics regarding cover.

Anything having to do with access will tend to be a bit messy because of the OSM tag source being the UK, which has cause a complex web of access-tied highway tags to develop that do not necessarily translate well.

(assuming we're talking about here)

The rendering of "highway=footway" in red is probably a historical artefact - footpaths on GB Ordnance Survey 1:50,000 maps are shown in red, and I suspect that it seemed like a good idea on OSM too (the same as with blue motorways).

If the OSM home page's "Standard" rendering isn't working for you you shouldn't need to change the data; just use a different renderer. It might be that the "Cycle Map" on OSM's home page works better for you (that shows them in brown rather than red, or maybe you might want to create your own maps based on OSM data.

There have been help questions in the past about it - if you search for "maperitive" (which is a way of creating your own maps that is often suggested) you'll find most of them. There are also numerous wiki pages - the "rendering" page is a bunch of links to get you started.

Finally if all you want to do is change the colour of "highway-footway" you could follow one of the sets of instructions on switch2osm - that describes setting up a standard OSM Mapnik stylesheet, but if you change "salmon" to "other colour of your choice" in there you'll get "highway-footway" rendered in that other colour.