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I never used a bow with stealth in Skyrim, even though I played as a pure combat mage (in plate armor, no less) and a melee warrior. In fact, despite the ineffectiveness of the one-handed weapon + one-handed spell combination (as well as the clunkiness of the game with this combo), Skyrim is one of those games where you can play as such an archetype without any particular concerns – the game isn't particularly difficult, after all.
As for a decent implementation of combat mages in computer games, Dragon's Dogma isn't so bad in this regard – the Mystic Knight not only has unique spells that no one else has access to, but some of them are directly tied to using weapons and shields. Although most of them are based on successfully blocking with a shield and counterattacking with different elemental effects. Magicka also allows you to play as a combat mage, albeit in a somewhat clumsy way, thanks to the ability to charge spells into a sword. Kingdoms of Amalur allows you to play as a hybrid character, as spells and weapon skills work equally well. However, the game itself is monotonous and quickly becomes boring. The Magus from Pathfinder is one of the most successful examples of this archetype, although the balance in the computer version encourages you to spend slots on buffs, rather than using the class's gimmick (which in the tabletop version allows you to do amazing things). Tales of Maj'Eyal also has several classes that combine swordplay and various forms of magic, although the game itself is very niche. You can find more examples, but I think these are enough for now. Although some of them are more successful than others.