Yes, you described how the system works. My critique is that ability scores and skill proficiencies don't have enough of an impact.
If you start bumping Arcana checks to DC 20 so that the clueless Barbarian with -1 can't succeed, the Wizard with +5 will also only have a 30% chance at it. A level 5 Wizard with Arcana proficiency and 18 Intelligence would have 40% chance. Hardly an expert.
It's not a good system. To make skilled characters actually skilled at something, the modifiers should be bigger and they should play more with stuff like Expertise i.e. double proficiency. The gap between the least skilled and most skilled isn't wide enough.
