TL;DR:
They simply picked an off the shelf connector readily available and fitting their specs (like size, reliability, stability, etc.).
Almost all of the mainstream classic desktop computers that had internal slots relied on the card-edge type of connector.
Not really. At least not when it came to (more) professional systems. I guess your benchmark would be early low-end micro computer systems here, hobbyist machines with a major design goal of being cheap. Edge connectors are about the lowest priced solution, with pin headers coming second. They are not reliable (short-term or long-term). Just remember how often "pull and plug" is the solution to problems with an Apple II or S-100 system. Card edge connectors miss reliable definition and implementation of many parameters.
Also, it might be useful to restrict comparison to other 16/32 bit micros of the mid 1980s. And no, neither the PC not the Amiga 2000 fit here due to special issues (see below).
would use Eurocard connectors for these slots.
They are not 'Eurocard' connectors, but 'VG-Leiste' as defined in DIN 41612. Eurocard only happened to use them as well, making it rather popular among computer users. The important thing is, as with other European standards, that it's a manufacturer-independent standard.
How come Apple came to use Eurocard connectors for the classic Macintosh?
Maybe because Motorola already used it for their VMEbus? VMEbus was introduced in 1981 (well before the Mac), developed from card edge based VERSAbus as part of becoming more professional.
In a similar way, Intel redefined Multibus using DIN 41612 connectors in the mid '80s when doing Multibus II.
In fact, even IEEE working group P1000 was planning to simply redefine S-100 using a VG-Leiste before settling for the more vendor-independent STEbus definition.
Essentially anyone designing a new bus connector after the mid 1980s was moving away from edge connectors. The AT bus doesn't count, as it was a kludge added to the PC bus, while AGP and PCI required quite sophisticated special type connectors, which not only added much strain on card construction in terms of quality, but also high startup cost.
VG in contrast is a reliable off-the-shelf solution offered by many manufacturers at competitive prices, already solving all issues new designs would have to take care of.
What advantage(s) were these style connectors seen to provide over lower cost and commonplace card edge connectors?
Reliable connection, defined wear for multiple plug/unplug cycles, defined forces, mechanical fit with less stress on PCBs, up/downward compatible layouts. Ability to hold cards without additional support.
VG connectors are always a good choice unless there's an already established standard connector for a specific interface that needs to be followed.
Seriously, the differences are so basic, I do not know where to start. Let's say: With a professional PoV, card edge connectors are redneck technology - much like using a calving rope to pull a trainer instead of a proper hitch - it might work well, but not really something to rely on.
Also, the cost point may be debatable, as even absolutely low end machines, like the 1984 Sinclair QL used a VG64. Contrary to popular belief, an edge connector isn't really lower priced than other types. While it is true that it comes (almost) free for the card, the motherboard side is at least equally priced with any other type. For a company producing the computer, a lower card price may not be of great concern. All motherboard connectors have to be bought upfront, while the card side is needed only if that additional card is sold.
Let's compare well established, multi-vendor types of similar size and pin count:
- A VG96 is ~7 Euro
- A PCI is ~6 Euro
- A PCIe is ~14 Euro
(Prices from Mouser Europe)
So despite being build in real huge numbers, the PCI connector is barely below a VG96, while a PCIe will be twice as expensive.
In fact, it might be less expensive compared to other alternatives - and what additional cost they carry:
Size is Cost
A 'classic' slot connector offering the same 96 pins would have had the size of an S100 edge connector, that is > 180 mm in length (*1) and ~15 mm width. A VG housing is 55 x 11 mm, occupying considerably less than ⅓ of an S100 type edge connector.
Motherboard size translates directly into cost. The same goes for interface cards, as the connector length defines the minimum length a card has to have, thus even the most simple interface card would have to be ~7 inches in length. Again, size is Cost.
Size is Size
While the Mac II is a comparable large machine, it's still way smaller than an S100 box. Looking at the mainboard

(Image taken from Wikipedia)
shows that an 18 cm connector might have barely fitted the board - if one can find a way to move chips, connectors and routing occupying that area somewhere else.
The Amiga 2000 shows the problems. With ~35x45 cm it's not only bigger than a Mac II board but considerable larger than a first generation IBM-AT board.
Now take a look at the way smaller Mac Color Classic:

(Image taken from this page)
I'm not sure how they could have placed a slot three times the size - not to mention an equal length card within the tiny cube Mac housing.
Stability
Edge connectors always need an additional way to fix them, as the slot does not give any lateral support. Either by a cage like with S100 or Multibus, or a slot bracket like the PC uses. VG does give inherent stability in all directions.
Form Factor I: Sheer Size
Even small cards with edge connectors will need additional support. This limits the ways a housing can be designed. That's the very reason most basic PCs still look like a variant of the classic huge box. Being less dependent on support allows a way greater variation of case designs.
Form Factor II: Orientation
With a slot connector a card has to be put straight in. A VG equipped card can be made the same way, or turned 90 degrees. A pizza-box Mac LC is only possible because of the ability to have PDS cards oriented that way (*2).
Routing and Vias
VG connectors bring all pins neatly in a 2.54 mm raster to the board(s), usually by through-hole. Any signal is automatically available on both sides of the board, simplifying rooting and saving many vias. Even more so when a board has more than two layers, as each of them can be reached directly.
For edge connectors, the same is only true on the motherboard side; on the edge side all signals come on one of two sides and only on the top most side, putting additional constrains on card design.
But Apple Could Have...
... bought a more dense edge design, or designed their own higher density variant. Sure. But why spend money (and time) to create something new when a more than fitting solution is available, off the shelf and eliminating many known issues? Not to mention that by being a type-specific creation that would tie them to a single supplier and high prices?
Some History
The VG standard was developed in 1971 as a German military standard as high density connector for anything from radio equipment and phone services to artillery control. It's a bit like the Sub-D connectors, as it primarily describes a shell that comes in various sizes, with clear definition about size and mounting points. Various definitions are available for type and number of pins. Beside basic 1 mm 2A connectors positioned in 2.54 mm raster there are for example also 3x1 mm blades for high current (5.5A). All pins are good for voltages up to 1000 V. It can come in up to 4 rows of up to 50 pins. Pin types can be mixed and so on.
It was intended for sturdy board-to-board connections, flat or as stacks, and board to rack. Thus intended to carry the full mechanical load as well. Variants are available for through-hole soldering, single wire soldering or wire wrap. It's available shielded, including compatible heavy duty metal 'outlets' for cables.
By 1976 the standard was published for non-military applications as DIN 41612 and quite well received for all kinds of professional applications from embedded systems to railroad or nuclear power plants.
Within computing equipment, the type C connector became the most popular. Its shell can hold one to three rows of 32 pins, offering great versatility especially for buses. It's been used for micros all the way to mainframes, replacing most vendor-specific connectors.
It's often taken as tied to Eurocard (100x160mm) PCBs, as that format was not only developed about the same time for the same purpose, but also became quite popular with modular computer systems.
*1 - ~16 cm pluggable length plus at least 2cm for bolts to hold them down
*2 - Yes, of course one could use a riser card holding another slot turned by 90° but that's quite an additional cost, complicating slot cards even more - not to mention the added stability issues. A look at some compact Compaq or Siemens PCs does reveal the unusually high mechanical effort necessary to cope with that.