I am not hiding skill modifiers in DCs....I am using DCs the same way they've always been used! It's DIFFICULTY CLASS. The DIFFICULTY of the task is determined by a number, called a CLASS. A windy and ice-slicked steep mountain sheer wall has a higher difficulty class to climb it than a calm dry inclined wall with lots of hand holds.
You're not saving any time by saying "All climbing is a DC 10" and then adding or subtracting a modifier for the check based on the difficulty - it's the same work, you're just splitting it out as a big modifier for the player to deal with. Just calculate the DC (like has been done in every version of the game), and have the player roll to see if they hit that DC. That's the d20 system - it's the entire point of the d20 system!
Below is what I would do.
DC Example Surface or Activity
0 A slope too steep to walk up, or a knotted rope with a wall to brace against.
5 A rope with a wall to brace against, or a knotted rope, or a rope affected by the rope trick spell, or a chimney or other area where you can brace against two opposite walls.
10 A surface with ledges to hold on to and stand on, such as a very rough wall or a ship’s rigging, or a corner where you can brace against perpendicular walls.
15 Any surface with adequate handholds and footholds (natural or artificial), such as a very rough natural rock surface or a tree, or an unknotted rope, or pulling yourself up when dangling by your hands.
20 An uneven surface with some narrow handholds and footholds, such as a typical wall in a dungeon or ruins.
25 A rough surface, such as a natural rock wall or a brick wall.
25 An overhang or ceiling with handholds but no footholds.
— A perfectly smooth, flat, vertical surface cannot be climbed.
These modifiers are cumulative; use any that apply.
+5 climb at speed in 1 minute
+0 climb at 1/4 speed
-5 climb at 1/2 speed
-10 climb at full speed
(for climbing kits I would have this listed in the equipment section and only talk about a climbing kit in general here)
+1 for masterwork climbing kit
-1 for poor or makeshift climbing kit
(I would have this be an application of these rules and not have them listed here or have them as examples of how the general rule is applied)
Disadvantage for unsafe conditions (slippery stormy etc.)
Advantage for safe conditions (rare but can imagine some magic items and spells, such as spider climb etc)
You fall on a fail by 5 or more. To catch yourself make a Dex save DC 20. To catch someone else falling next to or passing by you make a Dex save 20 and a Str save DC 20 to hold on to them. If you fail the Str save by 5 you both fall.
One other note about these DC, I am not clear on how they should listed, I have not come to a clear idea what their numbers should actually be. Is 25 a reasonable top DC? I am still unclear what the modifiers will be for characters and what they should look like in comparison to the DCs.