Describe it differently

The crux of this seems to be "Wonder Woman's Invisible Jet is silly," which is an issue with the fiction the rules inspire. I agree, it's a pretty silly thing to have to imagine. In this case, though, let's ask ourselves the question: do I have to imagine it that way?

We're talking about a beast that blends with darkness and shadow, not a wizard's Invisibility spell or a Predator cloaking device. If the whole problem is the idea of a halfling awkwardly floating above the ground, legs akimbo — well, imagine something else.

Describe it as "I'm riding a smokey blob of shadow," or "I'm riding a weird beast-silhouette with edges that waver like dark fire," or "I'm riding a regular ol' dog but its coat blends so well that you can't figure out where the dogs ends and the shadows begin til it's in full motion jumping out at you like RAWR." I promise nothing important will break if you say that you can't literally see through the big magic darkness-blending shadow-dog to the rider's leg on the other side.

Mechanically, you don't need to change much: the dog is still "invisible" as far as the rules are concerned, and the rider isn't. Feel free to describe being able to sorta make out the dog's shape or presence while it's got a rider, since just seeing the rider is enough for enemies to know its location and take a swing at it even if it's invisible. Feel free to describe how the rider is partially obscured by the dog's fur or its aura or whatever, but you can see most of them pretty clearly (that doesn't require special mechanics any more than "this broad-brimmed hat I'm wearing casts a shadow over my face" requires special mechanics).

There may be some small points where the mechanical and fictional description diverge, but in my experience that'll happen with detailed rules for seeing things in D&D across editions anyway. Just invoking the detailed concealment or invisibility rules has a tendency to bring out goofy edge cases.

(Experience: that's how we've treated shadow-creature shadow-invisibility whenever it's come up across different editions of D&D and it works pretty fine. Not transparent, just impossible to isolate from their environment.)

The most relevant parts of the find steed spell are (my emphasis):

... unusually intelligent, strong, and loyal steed, creating a long-lasting bond with it.

Your steed serves you as a mount, both in combat and out, and you have an instinctive bond with it that allows you to fight as a seamless unit.

While your steed is within 1 mile of you, you can communicate with it telepathically.

Telepathically is not defined in the rules so it takes its normal English meaning: "communication between minds by some means other than sensory perception." Importantly, "communication" is a two-way process: you can communicate with it and in can communicate with you.

In conjunction with "fight as a seamless unit," this means you can coordinate your actions with your steed better than you can coordinate them with your other party members. That is, anything you can do in conjunction with the rest of the party, you can do better with your steed.

As to the mechanics:

1. The mount has its own initiative (determined at the start of the combat irrespective of if it is being ridden or not q.v. 4).
2. The mount has its own movement, action, bonus action and reaction (q.v. 4).
3. The mount is "unusually intelligent": without direction, it will use its movement etc. in an "unusually intelligent" way. With direction, it will probably do what you telepathically say but it is an independent creature and, at the DM's behest, may do something different, subject to its "unusual loyalty".
4. When you mount your mount, you choose if you will control it or if, as it is "unusually intelligent", you will allow it to act independently as per the rules for mounted combat. In all cases that I can think of, allowing it to be independent is the superior option.