If your intention is to control Large and larger undead: that's pretty easy.
If your intention is to create Large and larger undead: you will either have to wait for WotC to produce the appropriate material, or... get creative with your DM's world.
A wizard of the school of necromancy is given little restriction on their use of their Command Undead feature other than the undead's Charisma and Intelligence (Player's Handbook, pg. 119):
Starting at 14th level, you can use magic to bring undead under your
control, even those created by other wizards. As an action, you can choose one
undead that you can see within 60 feet of you. That creature must make
a Charisma saving throw against your wizard spell save DC.
...
If the target has an Intelligence of 8 or higher, it has advantage on
the saving throw. If it fails the saving throw and has an
Intelligence of 12 or higher, it can repeat the saving throw at the end of every hour until it succeeds and breaks free.
Therefore, any undead you can see can be yours, as long as it is dumb and ugly enough and you are powerful enough.
Unfortunately, there are no spells or abilities published yet that allow creation of giant undead. However, if they exist in the world you inhabit, they must come from somewhere, right?
Therefore, your first task would be to find out what makes giant undead creatures in the world. Find out what is creating them and where:
- Is it an NPC necromancer who stumbled across an ancient artifact that can amplify the effects of necrotic spells (as in my campaign)?
- Is there a haunted graveyard where the restless dead can't help but stretch their legs? (common trope)
- Or is there a particularly unholy bog somewhere nearby that just imbues corpses in it with undeath?
Once you find that answer, simply harness that energy, adapt it to meet your needs, and you will be churning out giant undead slaves in no time.
Animate dead wouldn't work; the creature's actual type is unchanged.
The animate dead spell must be cast on "a pile of bones or a corpse of a Medium or Small humanoid within range", as you quoted - but the feign death spell doesn't turn a creature into a corpse (or a pile of bones).
Feign death says "the target appears dead to all outward inspection and to spells used to determine the target’s status." However, the animate dead spell is not a spell used to determine the target's status; it is simply a spell that only works on dead creatures. In other words, it is a prerequisite/requirement of the spell - not a function of the spell.
Because feign death doesn't change the actual status of the target, the target of feign death is not a valid target for the animate dead spell. One way to handle this situation is using the optional rule on invalid spell targets from Xanathar's Guide to Everything (p. 85):
If you cast a spell on someone or something that can’t be affected by the spell, nothing happens to that target, but if you used a spell slot to cast the spell, the slot is still expended. If the spell normally has no effect on a target that succeeds on a saving throw, the invalid target appears to have succeeded on its saving throw, even though it didn’t attempt one (giving no hint that the creature is in fact an invalid target). Otherwise, you perceive that the spell did nothing to the target.
Following this rule, nothing would happen to the target after animate dead was cast; the spell slot would be expended, but animate dead would fail. However, no indication is given as to why the spell failed; as Miniman pointed out in a comment on this answer, another reason it could fail is that it's under the effect of a gentle repose spell, for example.
Best Answer
From the description of the spell:
"[T]he creature's game statistics" includes the actions shortsword and shortbow. However, unless the skeleton has a shortsword /shortbow (and arrows) this is an action that is not available to them. They would need to be provided with these weapons.
In the absence of these, they can use the Attack action to make an unarmed strike. If you give them different weapons they can use the Attack action to attack with those, however, the DM would need to decide if they were proficient with them.
As for your assertion that a skeleton has an "an unlimited-ammo shortbow" - this is just wrong. From the Monster Manual p. 11: