This is looking at what appears to be a somewhat bizarre rules exploit, and attempting to determine if it is, in fact, viable under the rules.

Premise

We have a Bard who likes having friends - a lot of friends. He has Find Familiar. At level 10, he takes Find Steed and Find Greater Steed as his magical secrets and the DM has given him a ring of Spell storing.

Now, it's pretty clear that by himself, he can have one Familiar, one Greater Steed, and one Steed (perhaps a mastiff, because he wants a dog, and can't ride two things at once). We'll say that he chooses a hawk as his familiar.

Can the bard allow the creatures summoned by the spells to attune to and use the ring of spell storing containing Find/Familiar spells to summon Steeds/Familiars of their own?

A DM might adjudicate that a hawk/mastiff/griffin has no fingers and cannot use a ring, but Polymorph is available to a Bard of that level, and seems like it should be able to handle the issue for the Steed and Greater Steed. (I'm not sure if there are any beasts of low enough CR for the familiar that would have something appropriately finger-like.)

Is there a reason why the dog would not be able to have a perfectly loyal griffin of its own? Is there any real limit to the potential pyramid of perfect loyalty/obedience that would result?

This is in some ways similar to the linked question, and may have the same answer, but it is distinct on a couple of points.

• The first is that the linked question is about the Pact of the Chain
  familiar, most of which have obvious fingers already, as compared to
  the generic Familiar, Steed, and Greater Steed.


• 
  

  The second is that this question is specifically about the three Find
  spells, which seem like they might be a special case - it's not
  merely about whether your dog (polymorphed into an
  octopus/monkey/whatever) is able to cast the spell, but whether they
  are able to permanently maintain the found creature afterwards.

  
  
  

  In particular, this seems to severely break the intended limit of one steed, greater steed, and familiar per PC, and it seemed like there might be some further limit in place preventing it from working out this way.

  
  

All the way down (it works*)

As you've thought, there is nothing in this that creates any rules issues.

You've hit the pertinent points:

1. 
   

   Any creature can attune to the Ring of Spell Storing

   
   
   
   • They still need to spend the hour and be able to do so. But the 6 INT should allow for that for Steeds and commanding the familiar should work the same way.
   
   


2. 
   

   Whomever is attuned, including beasts, to the Ring of Spell Storing can cast a spell from it:

   
   
   

   
   

   While wearing this ring, you can cast any spell stored in it.

   
   

   
   


3. 
   

   Find Familiar, Find Steed, and Find Greater Steed are all 5th level spells or below and viable to be stored in the ring.

   
   
   

   
   

   Any creature can cast a spell of 1st through 5th level into the ring by touching the ring as the spell is cast.

   
   

   
   

This shows that your belief about this should working in fact does work.

There is one problem

You personally can't have Find Steed and Find Greater Steed going simultaneously. Find Greater Steed explicitly states:

You can’t have more than one mount bonded by this spell or find steed at the same time

However, that just limits your starting set to 2 creatures and they can do this again. You just don't start with three.

A DM's option

Having said that, the DM can still put the kaibosh on this if they feel it's taking away from the table or other players. One player having a huge array of creatures at their disposal may cause issues with other players to the point where it becomes unfun. That's going to be up to the table, the players, and the DM.

*The only issue to really consider is if the animals have the right appendage. It's important to note that if you do use polymorph to get around that, then the creatures will lose their intelligence and drop back down to whatever the new stat block says it is. A DM can rule that creatures of low enough intelligence aren't able to understand how to utilize the magic item or be able to maintain concentration for the casting durations.

In general, nothing in the rules prevent this, as NautArch's answer elaborates on, but the GM has quite a few points where he can overrule this method:

First, they have to agree that the companion can attune

Whether an animal can attune to a magic item is contingent on its ability to focus on the item, as discussed in this Q&A. Now as a familiar, you could command it to focus on the ring of spell storing, but its capacity to do so is up to the GM.

Second, the companion has to be able to cast the spell

Since all of find familiar, find steed, and find greater steed require Concentration to cast (as described in the Longer Casting Time rules), your GM may rule that the animals of only 6 intelligence cannot maintain concentration for the required time to cast the spell from the ring of spell storing. This is discussed briefly in this answer to a different question, but a GM may not be persuaded by the example-based argument.

Third, the new creatures have to be considered bound to the companions

Since each of find familiar, find steed, and find greater steed have a clause to the effect of...

You can't have more than one [companion] at a time.

... a GM could rule that any creatures created by one already bound to you is also bound to you by proxy and therefore would not take effect (so a familiar casting its own find familiar would just change form, and one casting find steed or find greater steed would simply fail.

One way a GM could handle it

While all of these are possible houserules a GM could employ to prevent such an exploit, if a GM were to allow it, there is an issue of bogging down gameplay for one player's companion army.

As a GM, I would instead group the collection of creatures into what amounts to a Swarm of Familiars, or a Swarm of Steeds. This homebrew monster could increase in size and power as the player dedicates more time to increasing the capacity for his army, but at least it wouldn't take too much IRL game time to handle the turns for the army.

I would also develop the majority of challenges so that vast quantities of companions is insufficient to succeed. Some of these include areas where a large number of creatures could not fit, and challenges that revolve around the quality of individual party members, rather than sheer numbers. Even still, I would allow for some challenges wherein an army of followers (however logistically challenged in the spreading of orders) is useful, so that the player doesn't feel that they've wasted their creativity and Magical Secrets.