Character mounts are currently just present in the game and offer relatively little strategic choice. What I mean by that is that you pretty much always end up taking the highest level mount. Why bother with a chariot when you have a dragon? Why would you ever use the Mechanical Steed on an Empire Engineer if you have the Steam Tank? The intended trade off is the upkeep cost, but from personal experience, that is never really a hindrance. This makes pretty much all low level mounts nothing more than a stepping stone. They are even granted automatically now when you achieve the required level. I find this to be a missed opportunity.
I think it would be interesting to allow for specializations on mounts. For example, you could either spend money or skill points (This was the case for Empire Captains in WH 1 and WH2 for Imperial Pegasus) on a Cold One mount instead of going for the dragon. I don't want to rival the max level mount in power in any capacity, but it could make if more interesting to let the lower level mounts gain utility that the max level cannot provide.
For example, chariots could get the engine ability from the Landship making it so that a character chariot could pierce through defensive lines. Archer Lords on horses could get extra ammunition or special ammo types. Cold One Mounts could get "Menacing Gaze" that would removed "Braced" status from enemy units in a radius. Eagles could cripple winged units to force a landing.  This would hopefully lead to some very interesting tactical maneuvers.
Lastly I would even recommend allowing for infantry specialization, not using a mount at all. A Mage at level 30 could get 3-6 ammunition for a projectile like Soul Cannon. A melee lord could get aura buffs, health upgrade or even be granted a body guard squad.
Really the sky is the limit, but the hope is that this would bring some changes to the late game. There really isn't much excitement in always picking the dragon.

I largely agree with you (on some mounts being obsolete and this being a genuine shame), but there are some interesting edge cases where this is already true, and the highest-level mount isn't objectively the best choice.

Take High Elf Nobles.  They're an odd example since their higher-level mounts have severe disadvantages (chariot loses the BvL and makes them worse at duelling, eagle loses the AP damage and has really weak attack animations), but I generally prefer them on horse or on foot.  They retain the AP and BvL when on horse, and foot makes them a small target.  I won't say I've never put them on chariot or bird, but it's not usually what I go for unless I absolutely need the niche that mount fills.

High Elf Princes are also a pretty interesting example - I'd actually consider the griffon mount competitive with, and arguably better than, the dragon alternatives.  It has fantastic speed, similar damage/melee stats, a smaller hitbox, and a very high attack speed, giving it very strong duelling capabilities for a monstrous unit, and making it very good at picking on enemy backlines in situations where a dragon might get shot down by enemy ranged.

I'm not entirely sold on the flying horse for Isabella von Carstein, either - I feel like the way vampires play, she performs better when she can embed into a block of infantry to have them tank for her, and not be susceptible to BvL.  Another interesting LL example is Sisters of Twilight - since their dragon is better in melee and you can summon the mount they're not using, you're actually incentivized to put them on the bird and summon the dragon.  That way they'll be faster, and you can send your strong melee monster into situations too dangerous to put your Lord in (and even sacrifice it if you want), while your archer Lord benefits from the higher speed of the eagle.

Several factions' mages also perform better on horse than X flying mount situationally, for the ability to hide in trees.