"The issue for me is that there are a lot more actions per day then skill ("spell") slots, all the way up through 20th."
"So it looks like there's a lot of cantrip use. The majority of casting will be cantrips until reaching 9th, and even high levels will be doing it some."
"The baseline we have from this is that casters will be mostly not-spells until double digits, and even at 20 will still have a good chunk of actions more than spell slots."
"Using mundane solutions also does not make casters feel magical."
"The idea of a few cantrips per day doesn't work - it still leaves mundane solutions for most actions until the highest of levels."
Here is my final idea. It does result in many more spell slots as level increases, of course, but they aren't guaranteed. There is a cost associated with it (bonus action) and you can't use it to recover spells cast as reactions or on someone else's turn.
Recover Spell Slot: As a bonus action, you can attempt to recover the spell slot of the spell you have just cast on your turn. Make a check using your spellcasting ability modifier plus your proficiency bonus. The DC equals 10 plus 3 for every level of the Spell Slot you wish to recover. (
Note: you can adjust the base DC and multiplier to get more or less slots, but for me these worked best and are represented in the table below. If you want to restrict this further, make it an action on the round after you cast the spell.)
The end result, at Tier 3 and especially 4, you have slots available even for the longest of days before your Long Rest. At lower levels, you are still going to use cantrips, but hey, you are just beginning, after all. The table shows the total revised number of slots you can expect by Level and Spell Level (checks begin at +4 for Level 1 and finish at +11 at Level 20).
This addresses having fewer slots than actions per day at higher levels. It doesn't make sense to have more slots than actions at lower levels IMO.
The majority of spells will no longer be cantrips (except at the lowest levels) unless you have an incredibly busy day LOL!
Depending on the number of encounters per day, even by 6th level you might have enough slots for all your actions.
There are not likely to be mundane actions as slots are more plentiful but not certain and come at a cost.
Cantrips are not reduced, just now there is the potential for many more slots, particularly at higher levels, granting more chances for utility spells and such.
Finally, you can adjust the concept as you see fit, giving more or less spell slots, that works best for your group.