In reality, the necklace of fireballs simply isn’t that good
Sure, it’s easier to use than a scroll or wand of fireball—though that tends not to matter very much to an adventuring party, since much of the party is going to be able to use wands, either as a result of their own spellcasting or because the UMD check is pretty easy to hit. Having low-level, untrained NPCs along with you to hand out the necklace to is just a logistical mess and unlikely to be effective, even if you can get past the issue whereby the followers provided by Leadership are not suicidal. Maybe a thrallherd could do it, but, uh. A thrallherd with a legion of suicide bombers is not a character I’d be comfortable playing with.
And sure, you can horde a whole bunch of them together, if you really want to, for one massive blast—expensive, though, and an eternal wand of explosive runes can do the same thing more cheaply. Harder to detonate, but usually that is a good thing when dealing with explosives—Alfred Nobel made the fortune the funds the Nobel Prizes because he invented TNT to replace nitroglycerin, not because TNT provided more force but because nitroglycerin explodes if you look at it too hard, while TNT you actually need to be trying to get it to explode. In the world of explosives chemistry, making new chemicals that blow up is easy—it’s making new chemicals that blow up only when you want them to that’s fiendishly difficult.
And all of these things are in comparison to a wand of fireball or an eternal wand of explosive runes—which are not good items themselves. Direct-damage spells tend to be rather difficult to use, requiring particular feats and class features to improve them because the base spells are on the weaker side. Otherwise, weapon attacks are usually better, and obviously those don’t use spell slots. The spells certainly can be used, of course, but when you put them in an item, you eliminate almost all of the options for improving them—even just increasing their caster level costs lots and lots of gold.
So in practice, all three of these items are pretty poor, and the necklace of fireballs is for most situations in most parties the worst of the three. You are almost-always better off just selling each of these and buying something more useful to you.
D&D 3.5e isn’t balanced and not everything has a real benefit
There are many, many, many options in this game that are strictly inferior to other options. Necklace of fireballs is a nice example, but there are more serious ones. I mean, consider the class tiers—this is not a balanced game. Any time you find an item or a feat or whatever else that just seems straight-up weaker than something similar, chances are it is simply weaker. It happens all the time. No harm in asking, of course, but it’s something to bear in mind.
Which means the necklace of fireballs is only really useful as a trap
If it’s in a pile of loot that is near where combat might happen, it might explode. If it is in a pile of loot and the players take it with them as they continue through the dungeon and get in more combats, it might explode again. It’s a trap—or, if you want to be generous, a gamble the players could take, can we keep this thing intact long enough to sell it, or is it going to blow up in our faces? But it seems to see most of its use as a kind of juvenile “gotcha!” gimmick for DMs who find that kind of thing “fun.”
They take damage from both fireballs
The key for me is the word "effects" from your quote:
The effects of the same spell cast multiple times...
I would argue that the damage dealt from the fireball is not an "effect" in the context of what your quote is talking about. This spell simply deals damage and has no additional "effects".
Furthermore (as pointed out by @PJRZ), the section you quote from begins with this sentence:
The effects of different spells add together while the durations of those spells overlap. (PHB, pg. 205)
This suggests that spells that simply have an instantaneous effect do not come under this rule, this they effectively have no durations. This also confirms that the context I mention above is for ongoing effects of a spell, where considering overlapping bonuses or additional effects makes more sense.
Instantaneous isn't a real duration, it's just a category of duration for the purposes of that property of a spell
Instantaneous means it has no duration; as for the duration of a spell being listed as instantaneous, I understand duration in this context to simply mean the name of a category, a property of the spell, but it isn't actually a "duration" unless a number is given (or "Until dispelled"); i.e. until a period of time passes to make "duration" meaningful.
From PHB, pg. 203:
Duration
A spell's duration is the length of time the spell persists. A duration can be expressed in rounds, minutes, hours or even years. Some spells specify that their effects last until the spells are dispelled or destroyed.
Instantaneous
Many spells are instantaneous. The spell harms, heals, creates or alters a creature or an object in a way that can't be dispelled, because the magic only exists for an instant.
Further, Xanathar's Guide to Everything (pg. 77) has this to say about Simultaneous Effects (thanks to @V2Blast for suggesting this):
If two or more things happen at the same time on a character or monster’s turn, the person at the game table — whether player or DM — who controls that creature decides the order in which those things happen. For example, if two effects occur at the end of a player character’s turn, the player decides which of the two effects happens first.
This also suggests that both fireballs should be resolved separately, making separate saving throws and damage rolls as though they did not happen simultaneously, but rather one after the other as though they happened a fraction of time apart.
Finally, the Essentials Kit includes some of these rules, but with a slightly different wording (thanks @Medix2). In the Essentials Kit Rulebook, on p. 3, under the section "Effects with the Same Name Don't Stack":
But when two or more effects have the same proper name, only one of them applies while the durations of the effects overlap (a duration is a time span of 1 round or more).
Durations is mentioned again, as in the PHB quote I included earlier in this answer (from p. 205), but in this case, what counts as a duration is made explicit; "a duration is a time span of 1 round or more". This also points to Instantaneous not being a real duration, further suggesting that both fireballs would deal damage.
Best Answer
You can throw a whole necklace for the equivalent of an "11th-level" Fireball
The description of the Necklace of Fireballs reads (DMG, p. 182):
This means you can throw a whole necklace for the equivalent of an "11th-level" Fireball.
This is the because of the specific mechanic of this item; it is an exception to a general rule (cf PHB p.7 under Specific Beats General). In most circumstances you cannot upcast a spell beyond 9th level, simply because there are no 10th or 11th-level slots.
However, in this case, if you were lucky enough to get a Necklace of Fireballs with 9 beads (1d6+3), and threw the whole necklace, you would be causing damage equal to an 11-th level Fireball spell.
Fireball is already a 3rd-level spell, so adding +8 beads brings the spell to an equivalent 11th-level spell.
The damage would be: 8d6 + 8d6, or an average of 56 points of fire damage on a failed Dexterity saving throw - which is not much, considering a 9th-level spell like Meteor Swarm for instance can potentially inflict 20d6 points of fire and 20d6 of bludgeoning damage, or 140 average on a failed saving throw.
As a side note, you can do much more damage by throwing one bead at a time! Throwing them this way, a necklace with 9 beads, can potentially cause 9 times 8d6 damage, which would be an average of 252 (9 times 28) damage on failed saving throws.