If I was remaking GURPS and keeping more-or-less the same structure, I'd start by combining a lot of skills. For one thing, any skill that's almost always used alongside another skill should just be the same skill – e.g. Electronics Operation (Medical) should just be part of rolls for Physician at the appropriate TL. I'd also default to broader skills – perhaps not to the point of having a "Science" skill like in Savage Worlds, but I could definitely live with having a single Economics that eats Accounting, Current Affairs (Business), Finance, Market Analysis, and Merchant. For broader skills, I'd include the option of subskills that would be significantly cheaper for use in a game that focuses on that particular thing. So, for most games Physician would be enough for a doctor/medic kind of character. But if you're doing, I don't know, House MD the RPG, then different medical specialties become much more relevant as a way to distinguish one doctor from another.GURPS has so many skills because it can be about anything. If I wanted to play a game about scientists and knowing which branches of science they are skilled in is important then GURPS has all the tools I need for that.
I'd also incorporate binary-type skills that cost like a point or two to get basic professional competency, and another point or two to become expert. In some cases, this would be used to expand one skill into something related – for example, letting you use your Musical Instrument (piano) skill to play the clarinet as well. But in many cases, it would just be "You know this stuff". This would cover things like profession-type skills, e.g. Seaman or Soldier, but expanded to other areas of the game. An IQ 10 guy driving a moving truck shouldn't need to spend 8 points to get Freight Handling up to even a measly 12 in order to know how best to tetris someone's stuff into the truck. Oh and that reminds me: who thought having skills max at 4 points per point was a good idea? Earlier editions had them max at 2 points for mental and 8 points for physical skills. 2 points for all skills seems more reasonable.
Another thing I'd do would be to make familiarity an explicitly optional rule, and perhaps only used in games that focus on that particular aspect. I'd also be much more generous with defaults, halving the current defaulting penalties as a starting point (so assuming Sleight of Hand and Filch remains separate skills, they'd default to one another at -2 instead of -5).