You point out, yourself:
The spell indicates that you can’t have more than one familiar at a time...
This is Rules as Written, as well as a clear indication at the designers intent. This is clearly not within the intent of the designers, even if you disagree with the why it doesn't work bellow.
Some DMs may reward the cleverness, others may see it as rule-lawyering rightfully (from a RAW and Rules As Intended point of view) dismiss the attempt. It is clear the designers want to limit people to one familiar from the find familiar spell:
You can't have more than one familiar at a time.
Note that it isn't tied to a source, or exception. You simply can't have more than one. That means for balance purposes the game designers decided you shouldn't have more than one. I mean, from a sheer action economy, you'd get the steed's turn, your turn, and two familiar's turns -- while everyone else sits and waits.
There is an Exception
The spell flock of familiars is a separate spell that contains an exception.
You temporarily summon three familiars – spirits that take animal forms of your choice. Each familiar uses the same rules and options for a familiar conjured by the Find Familiar spell. All the familiars conjured by this spell must be the same type of creature (celestials, fey, or fiends; your choice). If you already have a familiar conjured by the find familiar spell or similar means, then one fewer familiars are conjured by this spell.
D&D is based on the specific beats general. Note that the specific exception for more familiars is clear, and what the new limitations (1 hour duration, not a ritual, etc.) and changes to the base find familiar spell are layout. Nothing in find steed allows the casting of find familiars.
Moreover, this is in a module, which makes this new spell optional. The DM doesn't have to incorporate them into their campaign even if they playing the module.
The target doesn't have to be a creature:
A spell's description tells you whether the spell targets creatures, objects, or a point of origin for an area effect.
Find familiar doesn't target you at all. For instance no one would argue that fireball targets the finger of the caster despite reading:
A bright streak flashes from your pointing finger to a point you choose within range
We can all agree that it is the point in space that is the target. And magic missile isn't the caster despite beginning with the word "You". The target in this case is again, the people where the action happens:
You create three glowing darts of magical force.
The spell find familiar reads similar to magic missile and fireball in this regard. You do something, and something happens to or at a target. You have not changed - the world has, by the addition of the new creature:
You gain the service of a familiar, a spirit that takes an animal form you choose: bat, cat, crab, frog (toad), hawk, lizard, octopus, owl, poisonous snake, fish (quipper), rat, raven, sea horse, spider, or weasel. Appearing in an unoccupied space within range...
Leaving either the summoned animal form of the familiar or the space it appears in as the target (or both). Both are where the action of the spell actually takes place.
No Secret Rule
In a number of places you state "no secret rules" in D&D and I agree. Everything in this answer is quoting the source material all players and DMs have. Nothing is hidden. The target being the space where the summoning happens or the animal form that appears are both called out similar to other targets in other spells. What is also clear is that this spell has no exception to the one familiar rule.
Alright, your DM has ignored all the rules on targets and rewarded your idea with a second familiar. Who controls it?
As you are claiming you are target of the spell, then all the yous of the "second casting" would refer to the new target (the steed). So, all the yous would be replaced by the steed:
Your [The steeds] familiar acts independently of you [the steed], but it always obeys your [the steeds] commands.
In that case the DM would be in control (acts independently) but it would follow the steed's (who is under the DM's control [acts independently], but following your character's commands) commands to the best of its ability.