sorry - i do obviously disagree with you - but also there are a lot of inconsistencies with what you are arguing:
1. i may need some clarification on what "familiars typically do"–find familiar is a specific rule that creates a 0 CR beast that cannot attack (specifically note, "cannot attack" and not "cannot take the attack action" which are two different things). example every single familiar (which is still a CR 0 beast) can grapple (pending size of the target), which would be a general rule: grappling.
This was a reply to you. You stated
"familiars have specific DCs attached to their attacks." That statement is false. Further this thread is about familiars on a Warlock with Investment of the Chain Master which can both attack AND take the attack action per the specific rules of that Invocation.
Grappling is an unarmed strike which is an "attack" because the PHB says it is: "
In game terms, this is an Unarmed Strike—a melee attack that involves you using your body to damage, grapple, or shove a target within 5 feet of you." and it does not have a DC specific to a familiar, but rather uses a general rule for DC.
There are no specific attacks I know of for familiars that have a DC. This is certainly true for the Warlock-specific familiars, including the Skeleton this thread is based on. I believe it is true for ALL potential familiars in 5.5E. I could be wrong, you can scour the 5.5 monster manual to see if there is a CR 0 beast that has a DC on an attack, but I don't think there is.
The things familiars do that have specific DCs are not attacks. I named some of them - Heartsight, Sting and Scare from Sprite, Pseudodragon and Quasit respectively. If you look at 0 CR beasts you can add mimicry from the Raven. Those are the things with DCs that "familiars typically do" and they are all of the save DCs specific to a familiar that I know of.
3. the invocation does specifically state "If the familiar forces a creature to make a saving throw, it uses your spell save DC" and nowhere in that sentence does it specifically say "except when using and item" or "except when grappling" or "except when doing X"–nor does it say "If the familiar forces a creature to make a saving throw using one of its actions"–i am not sure what you are emphasizing with force, if the familiar's action requires that another creature make a saving throw, how is it not forced?
It does not say
"any DC that results from the familiar's action" it says
"any DC forced by the familiar" ... which to me means one of the familiars specific abilities or actions.
The net is forcing it not the familiar. Same thing if the familiar shoves an enemy into spirit guardians - it is the cleric's save DC, not the warlocks save. Same if it uses a wand on an enemy, it is the wand DC, same if he shoves him into Lava and he and he has to save to determine how much he gets burned These things are all "forced" by the item or environment, not by the familiar.
4a. yes, use warlock save DC to shove a creature up to 1 size category larger than the familiar, there is no mechanic for push, not sure what that is.
From the PHB, note the underlined and bolded:
Shove: The target must succeed on a Strength or Dexterity saving throw (it chooses which), or you either push it 5 feet away or cause it to have the Prone condition.
4c. spirit guardians specifically uses the cleric DC and is not an effect forced by the familiar (is this up for debate?)
Why would it when the Familiar is "forcing" the save by your definition?
I don't think it is up for debate, but it happening because the skeleton pushed the bad guy into the spirit guardians. So either the Skeleton forced it it or the Spirit guardians forced it and the same logic would apply if it threw a net.
4d. using an opposed skill check is not forcing a saving throw (is this up for debate?)
Not a skill check a DC - for example from Saltmarsh the Crumbling Bedroom:
Falling Through the Floor. The tracks left on the floor by the smugglers represent the only safe places to walk in the room. Each time a Medium or larger creature moves off this safe trail into the shaded area on the map, it must succeed on a DC 12 Dexterity saving throw or fall 10 feet through the floor, landing in area 6.
So skeleton pushes creature on to the floor do they make a Warlock save DC or a DC 12 Dexterity saving throw?