Well, unless corrected by the rw Captains, I'd say that your approach type depends on the equipment you have available, on the plane as well as on the ground.
When it comes to short ATC handling aspects, a visual approach may give you the most freedom of movement and also adds flexibility to the ATC's restrictions.

But I think your question was pointing at the 'which one may be the most appropriate?' aspect in the case of TNCM having all ground based stuff running.

Now they don't offer any precision approach procedure there, only some non-precision approaches are available.
The RNAV (GNSS) one is a LNAV +V type, so the vertical guidance only has an advisory character and you can only fly it down to LNAV minimums, which gives you a MDA of 700ft in our TNCM case.

The VOR approaches offer a minimum of 500ft MDA, so, while still being a non-precision one, we may call it more precise.
Since the 757 and 767 can't track a VOR radial with their AP modes, you may be using something way more precise on those occasions, which is the FMC itself.

As Peter pointed out, the database should give you a selection with waypoints and constrains, so you would remain in LNAV and follow those segments, while you can set up and monitor the raw VOR data on the FO's side.

So see yourself using the VOR Z Rwy 10 approach for example, your FMC has the approach loaded and LNAV is active, your Captain's side stays in the map mode and your FO's side monitors the raw data from the VOR.
In this mode you can descend down to 500ft altitude. The rest is done with the visual reference, so please leave the tourist guys at the beach intact.  8-)

The altitudes and distances are shown on the chart or come from the navdata, so VNAV may be an option or you just use the VS wheel to comply to the values.
This should all work nicely with the CS planes and also offers you two separate systems to monitor your progress.

Since you remain in LNAV for the lateral guidance, the FMC is completely independent of any VOR raw data. So even if TNCM turned that thing off, your FMC wouldn't be shocked at all.
Same goes the other way around, if your LNAV would become unreliable for whatever reason, the raw data can still be used.
So, either way, you may at least get close to the beach.  :)