Sure, but you have to be careful or you end up having to change the game so much you've basically created your own. As someone who used to make extensive houserules, I can assure you that can easily occur. Spells are ingrained into D&D; many problems and challenges exist in the game, starting from the very early levels, that only magic can solve.
The DM can make NPC spellcasters or workarounds exist, but by default, this isn't the case, because the game assumes someone is going to be able to cast the spell eventually. Players will get petrified, polymorphed, cursed, etc. etc.. There will be magical barriers and monsters with vulnerabilities non-casters simply can't exploit.
There will be enemies that the designer said to himself "this thing would be hard for a Fighter to deal with, but hey, it's not like he won't have a caster buddy to give him flight/water breathing/whatever".
WotC claims that you don't need casters to play D&D, but the reality is, that's only half true; the DM has to step in and either avoid magical challenges or give their players ways to overcome them in the form of magic items, consumables, boons, NPC's, or other such MacGuffins ("the saliva of a basilisk can be used to create a salve that undoes petrification").
So to remove the dependency of magic for high level play, you may need to redefine what high level play is, then hard code alternate ways to deal with all of these contingencies so that casters aren't necessary.
I haven't even gotten into the other things we rely on magic for, like crowd control and buffs. Obviously, martials would have to be redesigned to be able to perform as well as a spell can; right now, Battlemaster maneuvers and the like are far too conservative to do the job, likely to keep the warriors feeling "grounded" since, as has been noted many times, people like their warriors non-magical in nature.
And as for Improvise Action, you would have to put in actual guidelines for the DM to consider so as to make it a viable option. I'm sure most everyone has a DM story about a player who wanted to do something cool and, after being told to make several difficult checks, ended up with a worse result than if they had just attacked twice- this isn't bad or crummy DMing, it's a DM who, upon being surprised by the player, doesn't want to upset game balance or set precedent for something that can warp the game. I know I've been there ("oh you want to be able to make a called shot to blind the dragon? Yeah that's cool, roll with disadvantage"....three sessions later, every attack is a called shot) and I've seen it dozens of times ("you want to shoot the ropes on the ship to drop the sail on the bad guys? Uh, ok, so, it's your action, you need to make two attack rolls against fine targets....call it AC 25....then you have to make a Wisdom check at....DC 20 sounds good..."; and then the end result is two bad guys get knocked prone for a turn and are totally obscured until they escape the sails which they can totally do with their remaining movement).
Honestly, the simplest solution would just be to return magic items to the game full force, I'm talking AD&D style, characters can have scores of items for both offense, defense, and utility, none of this attunement nonsense.
I'm fairly convinced that the way 5e treats magic items is the reason players have to lean on magic use as much as they do, since WotC made them optional, but did nothing to replace what they did for the game in high level play (outside of some optional rewards in the DMG like boons just in case your DM thinks magic items are horrible, horrible things, lol). Actually it's worse than that- they made them optional and then didn't explain why one would want to add them to the game, so I'm sure there's many DM's who don't, and more that are very conservative about when they do.
Personally, I think every character should have the following by high levels:
*Magical weapon to harm foes (unless caster).
*Defensive items to boost saving throws and provide resistance/immunities.
*Items that provide the ability to fly, function underwater, scale difficult surfaces quickly, short range teleportation, improved stealth (invisibility, blending, chameleon power, cloak + boots of elvenkind), ability to survive without food/air/water, ability to see in darkness/fog/smoke, some way to safely rest in hostile terrain (Daern's Instant Fortress, Dimensional Knife, what have you).
But of course, 5e has taken almost every source of these things away from characters, leaving high level play almost completely reliant on either spellcasters or DM fiat.