So you are fine with non-magical healing. It's inspirational healing "forced" upon other characters that's the problem. Good to know so we don't waste time arguing for non-magical healing when that isn't your issue.
Sure. Spending HD on short rests, for example, is non-magical healing.
If somebody gives me back 20 HP as a bonus action, though, I do kind of want to know how...approximately...that works. If the ability is just written as "As a bonus action you can grant an ally five times your level in HP" with absolutely no fluff, then some of us can assume it's like Lay on Hands, and others can assume it's your magnificent, valorous 4th level Warlord shouting some spine back into my pathetic sniveling 18th level Barbarian.
In terms of flavor, if you want an option to describe what the Warlord is doing as somehow magical without his knowledge then I'm not opposed. This reminds me a lot of bardic inspiration. Is it magical or is it non-magical? But I do require the option of claiming all the Warlord's abilities are non-magical.
So does the flavor-less text version above work for you?
Dragon's fearful presence, Battlemaster's menacing maneuver. I would be perfectly fine with a DM ruling that a monster trying to intimidate me could give me the fear condition. If non-magical abilities can give me fear then surely a monster intimidating me could also give me fear.
I interpret a Dragon's fearful presence to be magical, albeit in the "supernatural" sense, not the spellcasting sense:
Each creature of the dragon's choice that is within 120 feet of the dragon and aware of it must succeed on a DC 16 Wisdom saving throw or become Frightened for 1 minute. A creature can repeat the saving throw at the end of each of its turns, ending the effect on itself on a success. If a creature's saving throw is successful or the effect ends for it, the creature is immune to the dragon's Frightful Presence for the next 24 hours.
It's not that the dragon is really scary to look at or roars terrifyingly (if it were, everybody in range would have to roll). The dragon just picks as many targets as it likes, skipping over others, and "poof" they are terrified. That just doesn't sound like mundane scariness to me. And if it were meant to be mundane, and Intimidate were meant to be used the way you suggest, why not just give the Dragon a huge skill bonus?
And the Battlemaster's menacing is for use on NPCs, not other player characters.
Speaking of intimidation, how would you expect an enemy trying to intimidate a PC work? If you get to decide your characters thoughts and feelings to the exclusion of all other game mechanics, how does intimidation ever work on a PC?
It doesn't, really. Intimidation/Deception/Persuasion are primarily for adjudicating the responses of NPCs, so that the DM has an objective way of resolving those interactions.
I might roll one of those abilities for an NPC and base my narration on the result, but I'm not going to tell a player, "Yeah...the Duke got a natural 20 on his persuade so you have to rescue his daughter from the mind flayers for free."