How does find traps interface with player skill focused play?

This is actually a great question, and part of the answer has to do with infinite play and what you wish to spend your time doing the other part has to do with the purpose of a trap.

The purpose of a trap is to make decisions meaningful. If there is no risk from just walking around, opening chests, or exploring rooms, then those activities become flat and dull. If the only threat is monsters, then you've removed uncertainty from the game making it flatter and less interesting. Traps should make the play of the game more interesting.

How?

Traps should represent an unseen threat. Lurking in chests and doors, they are likely not visible. But in rooms and corridors, there should always be some sign of their placement. This sign should not always be obvious, but should be visible to an alert party. I've written a whole lot about how to present this.

The other question becomes what do we want to actually be doing at the table? Do you want players to describe in detail how they are approaching and interacting with chests? The fact is, with any simple object like a door or chest, the ways of approaching and dealing with it are limited. It is possible to create an itemized list of activities to do when approaching a door. Depending on the type of campaign and your players this may or may not be an issue. It depends on what you want.

So, do I use the find traps skill?

I've found that it is hard to remember to do things. My general approach, is that all room and corridor traps are openly visible, and players either handle them manually in the game or suffer the consequences, as my traps and agency series. Openly visible means that there is some descriptive clue given to their purpose. I hid wall scythe blades in jagged shadows. Bodies on the floor is an option for easy detection, a side effect of trap placement in the description is more difficult to discern from the detection. For doors and chests, if they ask, I will simply tell them if the door or chest is trapped.

They frequently don't ask.

It is apparently really really difficult to remember to do that. 

This works very well to put the focus squarely on player skill. I want to point out, that I only do this for searching for traps, not most activities, like packing, lighting torches, etc. Specifically, I don't assume players will hit in combat, a key pillar of play any more than I assume they are going to take the time to look for traps another key pillar of play. Standard D&D does not consider remembering to pack your sword as a pillar of play. There are games that do. Again, it depends on where you want to spend your time.

Most of the games I run at this point, don't even contain a search or find traps skill. The idea behind the original thief skills, was that they were semi-magical in nature. So in systems that do have a find traps skill, I generally run it as either a Danger Sense skill, or allow it to be rolled as an additional saving throw to avoid the effects of traps. The same with locks, in most campaigns - any normal lock can be opened by a thief*. It is puzzle locks, difficult vault locks, and other special tricks that Open Locks is expected to address.

*Numenhalla, my megadungeon campaign, is an exception. The focus is on dungeon exploration and locks and the environment are a large part of that.