I am extremely ignorant when it comes to these matters, but just for curiosity: why do we need to reverse-engineer the original hardware? (..)or is it the case that the internals of the different chips that made up the system (e.g., the VDP) is the part that is unknown?
Yes - it's the internals of chips that require reverse engineering. In the case of MSX, basically all done except for the V9938/V9958. (that's mostly there, but w.i.p.)
I just wonder whether it would be possible to build a new MSX machine today by just buying a new Z80, RAM, etc.
Yes that works fine. Required IC's can mostly be found on eBay, specialized webshops etc. No reverse engineering needed, 'just' wire the stuff up & put suitable ROMs in it.
As for FPGA's: it's a matter of availability / cost / size / power consumption & flexibility. Original IC's are old & thus prone to breakage, prices may vary, stocks may be limited, etc, etc. For the most part, an FPGA re-implementation is smaller, cheaper & lower-power than using the original IC. And the additional cost of yet-another-extension is near 0. So it makes sense to pick a big enough FPGA & implement everything in that. If a bug is found, a new version of a softcore can usually fix the issue. Easier than re-wiring a board...
The high price of boards like this is a numbers game, really. If 100k+ units were built, they would be cheap. But with a few hundred or thousand units, development costs are a large part of the equation. Last time I checked, price of a '1chipMSX-capable' FPGA was around $30. But then you need to put it on a board. And design that. And have that board produced. And connectors. And on-board support circuitry (power, memory, ...). And a case. And develop an FPGA configuration for that board. And test it all. And put it in a nice box. And ship those. And replace defective ones. And do so for 100 or 1000+ of them. That's a long route & it all adds up... 
Quite a few KickStarter projects have failed because people underestimate the difficulty of prototype -> mass product.
I think it's more subtle. You won't get the original hardware on the FPGA. But you may get the same or very similar behaviour. That does require to reverse engineer the chip in full detail. Then again, the same thing must be done for a software emulator and it will get you the same results...
Only with sw emulation you'll suffer from stuff like input lag. Can be a deal breaker if you're serious about gaming (notice e.g. the difference between a shmup in regular MAME vs Shmupmame).
(Interested if openMSX can be optimized that way, at the cost perhaps of display accuracy?) (Though I'm not sure how much openMSX suffers from it? A while ago I did have the impression that openMSX felt more direct than blueMSX playing Shmup!. No idea how it feels on real hw though.. (I am interested to know if that's (rather) noticeable? 
))
Hm. Been playing to many bad Spectrum->MSX ports to even click that link.. 
Hm. Been playing to many bad Spectrum->MSX ports to even click that link..
Oooooh... Not completely false though 
Let's not forget that this project could give birth to an MSX version if succeeding (as MSX user Fabio Belavenuto is involved here)!
The case looks awesome imo, and it being FPGA it may even be possible to port the ese3/1ChipMSX core to it (Though I dont know the size of the fpga used)