Why can a flugelhorn easily play its fundamental frequency, when a trumpet can't?

In a simplified model, the tube is effectively closed by the player's lips at the mouthpiece end. For the theoretical case of a straight tube with completely cylindrical or conical bore, and no bell, the harmonics of the conical bore have frequency ratios 1:2:3:4.... but the cylindrical bore only has the odd harmonics 1:3:5:7....

Those facts are most obvious in woodwind reed instruments. The clarinet has a cylindrical bore and the first harmonic is at the 12th (3 times the fundamental frequency) but almost all other instruments (e.g. the saxophone) have conical bores and the first harmonic is at the octave (2 times the fundamental).

It is also apparent in organ reed pipes, where the pipes designed to imitate trumpet tone are literally conical along their entire length, with no bell.

In "cylindrical bore" brass instruments, the large bell reduces the frequency of each harmonic compared with the simple theory, so the actual frequencies change from 1:3:5:7:... to something close to (a number less than 1):2:4:6:... The actual shape of the "cylindrical" tube is carefully designed to make the higher ratios as close as possible to 4:6:8:... and so far as the player is concerned, these are equivalent to the 2:3:4:... of a conical tube of twice the length. Hence the description of "half tube" instruments which are (approximately) half the length of a "whole tube" instrument that from the players point of view sounds the same harmonics.

However the fundamental frequency can not be accurately "modified" in the same way as the higher harmonics. It is always physically playable, but it may be so out of tune, and its frequency very unstable and depending on the exact tension in the player's lips when blowing, that it is musically useless and hence described as "unplayable" in practice. The embouchure for playing the fundamental may be quite different from playing the higher harmonics, so there is no practical use for learning how to play a note that is completely out of tune in any case!

It's worth noting that the physicist Helmholz who founded the modern study of acoustics in the 19th century got his theory of brass instruments completely wrong, because he didn't realize that the player's lips effectively formed a closed end to the tube and not an open end. And his reputation as a scientist was so great that it was half a century or more before anyone seriously questioned his incorrect theory! Of course musical instrument makers had been constructing practical instruments for centuries before that, and musicians had been playing them, with no theory at all except the empirical observation of the frequencies of the different harmonics.

Yes, it is the conical bore which makes that the boundary conditions near the mouth piece are like those for an open-open cylinder. Here is an explanation with many links.

I like to think of it as the mouth piece in free space, in the center of an imaginary sphere. Now put some imaginary radial conical walls in this sphere. Those won't affect the longitudinal motion of the air in spherical waves from a point source.

I encountered this when I tried to devise a course lab experiment with a plastic vuvuzela trumpet and got initially puzzling results...

I started on trumpet more than fifty years ago, and have played all the brass in school bands. I can play the fundamental, but of course there are unreachable notes between fundamental with all valves up and second harmonic with all down.

By the way, a trumpet’s second with all up is B♭, not middle C, and with all down, the E below that.

Fundamental is more difficult, but whether the tone merits the adjective “clearly” is opinion-based. In fact, after listening to the Marsalis solo cited in another answer, I would say he played the fundamental very clearly.