If armed combat and skill use at one table "boils down" to the ability to roll high enough on a die, but at another table armed combat and skill use "boils down" to something else (e.g. the players' in-combat IC choices, or the players' out-of-combat IC choices, or the players' OOC build choices, or the DM's adventure design choices, etc.) then the experience of the martial/caster divide is inescapably going to vary from table to table.
The mechanics of combat and skill checks in 5e
are the d20. In combat, it's all pretty defined, there's a limited number of options, they're resolved with d20 checks, AC and such are known monster quantities, decisions made in combat may give you advantage or disadvantage, there's the added factor of a damage roll and that's about it. With Skills, there's no initiative order or damage rolls, but you can declare an action that the DM may simply deem successful, so that's something to work towards to avoid resting it all on the d20.
So, yes, regardless of table variance, the system boils combat and skills down to d20 checks. The DM may override the system, is essentially encouraged to bypass it as a matter of course in the case of skills, but that's what the system is. Roll a d20 + bonuses vs a DC. Indeed, that's D&D/d20's claim to rules-lite simplicity. That's why it seems so hard for the designers to 'buff' martials in some way other than bigger numbers - because they're ultimately just numbers.
Spell, OTOH, do what they say they do. That may be an attack roll and damage roll. It may be a saving throw. It may be a damage roll with a save for half. It may be just a damage roll. It may be text describing something that just plain happens every time. And, depending on what you declare you're doing with it, it may accomplish something beyond that in the DM's judgement, too.