The synthesist gets nothing from the eidolon's saves, and only part of its ability scores.

From the PFSRD:

Fused Eidolon
...
While fused with his eidolon, the synthesist uses the eidolon’s physical ability scores (Strength, Dexterity, and Constitution), but retains his own mental ability scores (Intelligence, Wisdom, and Charisma). ... The synthesist uses the eidolon’s base attack bonus, and gains the eidolon’s armor and natural armor bonuses and modifiers to ability scores.

There's nothing in there about saves. If something gives the eidolon a bonus to saves (such as the progression eidolons get as the summoner gains levels), the summoner gains no benefit from it. It doesn't matter which saves are the eidolon's good saves: the synthesist does not get those bonuses.

A synthesist will tend to have higher saves than other summoners though, thanks to ability scores. Because she uses her eidolon's Dexterity and Constitution ability scores, she will use its modifiers in those abilities rather than her own (which are likely to be lower). This will tend to give her higher reflex and fortitude saves than she would otherwise have.

Shielded Meld (which kicks in at level 4, incidentally, not level 2) does improve the synthesist's saves, because it explicitly says it does.

Thus, a level 4 summoner would calculate her saves as follows (while fused with her eidolon):

• Fortitude: +1(summoner) +C(eidolon's Con modifier) +2(shielded meld)
• Reflex: +1(summoner) +D(eidolon's Dex modifier) +2(shielded meld)
• Will: +4(summoner) +W(synthesist's Wis modifer) +2(shielded meld)

(Plus any other bonuses, such as from magic items, feats, traits, etc.)

TLDR: Synthesists are already ridiculously broken; don't make it worse by giving them bonuses to saves they're not supposed to be getting.

The phrase means casting a spell from the item

This effect applies whether you are using the instrument as the source of the spell or as a spell casting focus.

Using an item as the source of a spell can really only mean using the item like a wand or rod that contains a spell. There is simply no other mechanical meaning it can have. No other rule or mechanic enables casting from the item in a way that does not involve using the item as an arcane focus or casting a spell that is contained within the item (using the item as a spell's source).

The Doss Lute can cast a spell that causes the charmed condition

You are not entirely correct when you say "none of them have spells that charm" though. The Doss Lute has the spell animal friendship which says:

...the beast must succeed on a Wisdom saving throw or be charmed by you for the spell's duration.

But that is indeed the only spell that causes the charmed condition out of all the instruments of the bards.

In fact, the only other spell period that has a charm effect and a material component is hypnotic pattern.¹

As to why the designers did it this way, we can only speculate. It seems likely that it was an oversight and that the items were intended to cast more charm spells at one point in development. Or it could just be future proofing for further instruments to be added later (potentially with more spells that cause the charmed condition).

^{1 - See What spells can an Instrument of the Bards actually impose disadvantage on? for more on the background of this issue.}