Evolving from Wargaming, the earliest editions of D&D were heavily into combat capabilities first and everything else second. There were rules for picking locks etc, but initially at least they were a totally different mechanic and felt tacked on. Spells were the primary means of giving non-combat powers and abilities, but skills were [subsequently] greatly expanded and better graduated. But it still fell well short of all the other roleplaying systems based around a vast array of skills and capabilities - Call of Cthulhu, Runequest, Dr Who, Traveller, and so on.
But does having a raft of skills and capabilities reduce or enhance roleplaying? In some games, particularly Sci-fi or Modern games, you need these skills. How good your PC is at piloting a spaceship or his skill at hacking government computers is critical to the story, so you will need structured rules for these situations. You also need a way to differentiate [for example] an expert pilot from a merely good one. But in a pseudo medieval world the challenges you are overcoming are [on the whole] physical ones requiring combat.
At its core I guess every game needs challenges to overcome and it's the nature of those challenges that dictate whether you use skills and abilities or combat rules to overcome them. In a game such as Call of Cthulhu where combat is definitely a last resort you need a raft of skills to give your players the ability to overcome obstacles. In the more combat orientated D&D world you utilise detailed combat rules to progress.
A good game is one where you are not [relying solely on] die rolls to determine your success. Similarly in a combat-based RPG you want a game that allows you to select tactics and weapons to enhance your success rather than a never ending series of to-hit and damage rolls.
All of which raises another, possibly more important question. Is it better to resolve social interactions with dice rolls or should they be role-played? Do you allow your players to use their social skills, or limit them according to the stats their characters possess?