Sounds like they followed the standard fab steps according to your design.
The error, then, lies between the user and keyboard, as the saying goes. Not that blame does any good after the fact, but we can take a moment to understand what went wrong between expectation and reality.
I'm assuming you expected a board just like in FR-4, with insulated PTH (plated through hole), annular rings, and all that.
Basically, the catch is: FR-4 is an insulator, so it can simply be drilled, chemically activated, and plated. To do PTH on metal-core, additional hole-filling steps are required, and that adds process cost. It seems they do not do this in the process you ordered, and they substituted NPTH (simple drilled holes) instead.
Not so much that the holes are conductive inside, then; more that they were never intended to fit component leads. You would never be able to solder them in place, after all -- no plated 'barrel' to wick solder up the lead and support it.
How could we figure this out beforehand?
Look up the process and stackup at the manufacturer's website. If they don't document it, or you are left with questions: ask! Sales is generally responsive. If that's still not sufficient, keep shopping; there are hundreds of PCB manufacturers to choose from. Unfortunately, many don't document their process, or in enough detail to really design to on a first-time basis; maybe they make industry standards, but if one is not familiar with them, that isn't much help. (But, now you are familiar, with one example anyway!)
Mind, you may end up paying a premium for a well-defined fab process -- even if the result is fairly pedestrian, like the single-layer NPTH you received. Not that it's any help to a small-timer or hobbyist, but this is just the cost of business -- the reality is, a couple hundred bucks here or there is throw-away money when a corporation has multiple employees waiting around for parts to arrive. I've signed off on orders from $200 to $6000+ for PCBs and PCBAs -- and those are very much on the small, low-complexity end of things!
What can you do with the fab?
Well, now that you know what an example looks like -- and, the most common construction I've seen with these, is single-sided designs with surface-mount components. Typically for LEDs or power transistors, and few support components. Simple circuits/layouts that don't need a lot of (or any, hopefully) trace crossings -- with a single layer, you have to make do with jumper components ("zero ohm" resistors, usually), spending still more layout area and BOM cost. Very simple circuits like strings of LEDs avoid these costs.
And keep in mind, you can always split up the design. As long as you can get everything heat-dissipating on the metal-core board that needs it, and interface the rest elsewhere -- then you can always add headers or whatever, and attach a regular FR-4 board to it. This is a convenient way to handle the power, while still needing the complexity of, say, a microcontroller plus communications peripherals, something like that.
What else is out there?
Disclaimer, I've not surveyed the market, nor shopped or bought anything relevant here. I do see some hits, at least, for multilayer and PTH metal-core PCBs. How much cost these processes add, you'll have to get a quote to find out.
I wouldn't be shocked if such a build finds quotes from 10s to 1000s of USD, depending on where the manufacturer is, and what kind of process they're tooled up for -- costs come down rapidly with quantity, but if you aren't ordering at least hundreds at a time, to them you're basically doing a special one-off prototype order, and the setup time, masks and labor costs are dominant. Asian proto services can offer prices towards the low end of that scale, but might still hit the middle (or more) if it's more of a full-custom run for them.
Designing
As for how else to design what you've ordered -- offhand, it looks like some terminal blocks or other connectors, maybe a capacitor, polyfuse or MOV, transistors or headers, etc.
The connectors are the most ponderous to substitute: SMT is notorious for poor strength -- not that it's outright bad on typical materials, I would even say it's impressively good, considering it's resin-bonded foils and solder joints -- but connectors just expect so much, especially when a heavy cable has leverage against the poor connector (which is very likely the case for terminal blocks!). Possibly, you can find a similar type but which offers board-lock pins (plastic pins that lodge into NPTH (or PTH, doesn't matter) holes in the board), or that has ears which can be bolted down, etc. There are also types which panel-mount in the front, taking the bulk of connector stress off the board (but the board itself needs to be mounted well to the panel/enclosure to avoid stressing the solder joints!).
It's even an option to just tack-solder wires directly to the board, on wide SMT pads. This provides no strain relief whatsoever, of course, but it does invite the designer to find a solution to it: you might add holes or slots to snake wires or tie-wraps through; apply glue to anchor the wires; snap or screw the board into a bracket or enclosure that provides strain relief; etc.
If you are using transistors or other semiconductors, D(2)PAK, DFN-5x6 and other packages offer fantastic performance -- these are normally rather mediocre on fiberglass, with power dissipation ratings in the 1-10W being typical, but with the thin layer of insulation over a freakin' solid chunk of metal -- thermal performance can be quite excellent here, the board acting as a heat spreader, allowing heatsinks to be placed at some distance as well. I wouldn't want to be pushing multiple 10s of W from such devices anyway, but ~10W will be easy, and without any of the song-and-dance that you need on a fiberglass board (thermal vias, multilayer pours, etc.).
And for switching application, the board provides a dense ground plane, confining currents close to traces, minimizing stray inductance.
Other components can be substituted similarly. Of note, PTC fuses are not such an example, however: because they depend upon heating up to function, putting the generous heatsinking of an entire plate of metal underneath them will significantly raise trip current, and delay operation. (This is even a problem for fiberglass boards, where it can be desirable to avoid inner planes beneath PTCs.)
If you're inclined to object to SMT because of precision placement, or hard-to-see parts, I would say it's not as bad as you imagine it; just take it slow and feel out what you can do, what you can see, and where you need tools to help place things, or magnifiers or other vision assistance. (For my part, my eyes are still pretty good, but I regularly inspect things with a 10x loupe just because it's so much faster to look at things up close that way. I'll probably get one of those heads-up magnifier things one of these days; or maybe a desktop/overhead magnifier camera.)
Also, metal-core boards have the distinction of being nigh impossible to solder by iron. You will need preheating at least, and doing hot-air reflow soldering (whether with solder paste, or pre-tinning the pads and adding paste flux) just does everything in a single step. Be patient and wait for the board to come up to temperature; don't rush it or you'll burn components and flux.
