Not really, because of the limitations of proficiency and non-skill checks
Because we're used to level 15 mages raining fire and it's all on paper, we tend to lose the gravitas of how powerful a level one character really is compared to the average joe, and things like proficiency tend to be understated.
Proficiency comes from practice, spending days and years doing the same thing, learning the ins and outs of what and when. Skills are things that can be practiced, and thus it is fairly common to have proficiency in a handful of skills over a lifetime. You can even become proficient in tools over the course of a year.
But to get an idea of how difficult proficiency really is, there is no in-game way to train yourself to proficiency with a skill because that kind of training takes years. You can't even train yourself to be proficient in a weapon without adding a player class, and most characters only have one player class in their whole life.
But checks like memory and initiative rely on things that can't be trained at all. You can't practice yourself to be more alert or have a keen memory. You can't even practice yourself to be proficient in resilient mental defenses or abnormal weapons. Improvements on these rely on feats and racial abilities, or in role-playing terms, innate talent.
You can train your abilities to compete with those talented (insert anime montage here), but you cannot be proficient in your talents. Experience and expertise just don't play a role in acquiring these abnormal abilities.
Initiative especially is tricky from a meta perspective because it's supposed to represent all the chaotic little factors that make people slower to react and faster to get moving. And when I say little factors, we're talking about splitting 6 seconds (a round) down into 1d20 pieces, so the +5 granted from alert is only really helping someone move a second faster.
From a gameplay perspective, it's really not good for battles to become predictable, and one of the easiest ways to shake things up is to have people going in different orders in each combat. The less random that initiative becomes, the more predictable battles become.
TLDR: You can have advantage on these kinds of checks, you can have flat bonuses to these kinds of checks, you can improve your abilities to benefit these checks, but proficiency and non-skill checks aren't supposed to overlap.