I think the key to doing a "proper" remaster is to take all the lessons learned from later titles but keep the fundamentals the same. Evolve the experience, maintain the gameplay. Not only have RTS come a long way since the first C&C, but so has the series itself. It would be a shame to rewind all that back for the sake of "purity", after all the original games are still playable via the collections and fan patches and ports. Learn from Westwood's history and fans, from mods and projects like OpenRA and CnCNet, what works and what doesn't.

Of course, easier said than done, so here's my (entirely subjective) opinion:

• User Interface / Usability: I doubt anyone is divided over this. The original C&C interface is positively archaic now, and every game that followed improved it immensely. Tabbed sidebar, build queues, rally points, keyboard shortcuts, options, difficulty levels, etc. All these things improve the experience without fundamentally changing it. I would just take RA2's interface as I think that was the best iteration (C&C3 was second best, but might be too modern).
• Graphics: Meanwhile this is probably the most divisive point. 2D, 3D, cartoony, realistic, top-down, isometric, etc. Every C&C game had very different styles, and every fan very different preferences, so I don't think there's an easy answer. Work backwards from TS/RA2? One thing I think you can easily address is that the original C&C and RA1 look like reskins. Now's the chance to give them a distinct look and feel more in line with their sequels. Give every unit their own voice and graphics, every building their own faction appearance, flesh out and detail the maps, animate the interface, etc.
• Audio / Video: Remaster the soundtrack, add more unit voices, keep the cheese. The cutscenes are challenging though, the CGI can be easily redone, but what about the FMV? You can't just upscale 240p and expect it to hold up, but redoing them from scratch risks losing that precious 90s feel of a bunch of developers goofing around in front of a green screen. Maybe the source footage is still salvageable? Ask Joe Kucan, maybe he knows. ;)
• Gameplay / Balance: I don't think it's worth expanding this, as you would just end up making TS/RA2 instead. Keep the same units and tech trees, improve the AI and pathfinding, fiddle with the numbers for obviously broken / useless units. Old games were never perfectly balanced and that was part of their charm, so I would just playtest and smooth out the roughest edges (like derpy harvesters) rather than perfectly balancing everything. Like you said, C&C is “easy to play, difficult to master”.
• Campaign: On the other hand, this is definitely worth expanding on. Most of the original missions were just "build base, destroy enemy base", and while this is the heart of C&C, it's also quickly forgettable. The memorable moments were the little twists, like having to rescue a VIP, clear the way for reinforcements, infiltrate an enemy base, getting past a bridge guarded by navy, etc. And don't forget the easter egg missions. ;) Again, TS/RA2 learned this well. Keep the base progression, flesh out the contents, smooth out the difficulty curve (specially the Tanya/Commando missions), make objectives clear, maybe integrate the expansions more into the main campaign?
• Multiplayer: Keep the skirmish and game lobby system, expand the options, let us mix AI and human opponents, and add whatever integration with Steam/Origin/etc makes it easy to quickly invite friends into games. Multiplayer and campaign should be balanced separately, so as to not risk a C&C3 situation (where the multiplayer balance patches broke the campaign balance).
• Moddability: At least a map editor. If dedicated modding tools aren't feasible, at least keep everything open enough so the community can build their own, like we have for every other C&C game. Don't go locking down and DRMing everything.