This is completely untrue. 5e's skill system looks nothing like 3e, setting aside the question of skill points entirely. 3e skill entries start with a complete description of what each skill allows, followed by a table of modifiers, followed by any modifying kinds of checks (climbing 1-handed, for example). 5e has nothing like:
Instead we get a fun note buried in the Move action, that specifies sometimes your DM might ask you to make an Athletics check as part of movement, or they might not.
After all that, 3e skills specify the time it takes for each attempt, whether or not you can take 10/20 (a thing you were frequently expected to do, because the results of getting specific number were spelled out above), and then guidance on whether you can try again. Hard to compare to 5e's comparatively quite abbreviated skill section, and absolutely opposite in orientation. Given a description of a situation, skills in 3e simply don't offer choices to the DM about adjudication; you don't set DCs in 3e, you derive them.
You're correct in identifying the 5e skill system is largely taken from 4e; all it's removed is the table 42 set of difficulties by level in favor of a single band of difficulties. In play, it's got far more to do with skill challenges than 3e's skills; you propose a course of action, the DM decides a difficulty with some guidance, you succeed or fail. The difference is entirely down to the number of checks involved before the result is determined.
The most uniquely 5e skill design choice is automatic pass/fail on natural 1/20s.