Answer
Can I remove only the GPS location and keep the rest of the metadata?
Yes. GPS is stored in its own section, so it can be removed on its own while camera settings, copyright and timestamps stay untouched.
Updated
Location lives in a dedicated structure — a separate directory within the EXIF block, holding latitude, longitude, altitude, a bearing and a timestamp for the fix. Because it is self-contained, it can be removed without disturbing the fields around it.
That matters more than it sounds, because for many people the metadata is worth keeping. Photographers rely on exposure settings, lens data and capture times to organise and edit libraries. Anyone publishing work has a real interest in keeping the copyright and creator fields, which are the machine-readable statement of authorship. Stripping the file to nothing throws all of that away to solve a problem that was confined to one section.
The trade is that a partly-cleaned file still identifies your equipment. Camera make, model and — on many bodies — the serial number remain, and a serial number links every photo from that camera together. Whether that matters depends on what you are protecting against: it is irrelevant if the concern is a stranger finding your house, and central if the concern is two anonymous accounts being connected to each other.
The GPS location remover does the narrow version. The full remover takes everything, and the viewer will show you exactly which fields are present before you decide which of the two you want.
More on this: GPS location remover.
Terms used here: EXIF.