I of course agree with you on what the mechanics of to-hit rolls and skills checks are. I don't agree with you, however, that combat and skills necessarily "boil down" to application of those mechanics.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.
Combat in 5e can be about trying to attack and deplete the enemy's HP before they deplete yours, but it doesn't have to be. For instance, if at a particular table the outcome of combat is rarely in doubt, then combat at that table is more about who the PCs fight, when, and why, rather than about d20 rolls. Examples include games where the challenge comes from trying to stack the odds in the party's favor prior to combat, or old-school-style games where the challenge comes from trying to avoid combat, or games where the players are skilled at having their characters seize the strategic initiative, or games where the DM frequently designs combat encounters with goals other than killing all the enemies.
Each of those tables will experience the martial/caster divide differently from tables where combat does boil down to application of the combat mechanics to see who runs out of HP first.