This pronunciation is not the only one; here the normal pronunciation is retained.

I

I will offer an explanation on the ground of the respective practicallity of the two possibilities, as I perceive them. From the point of view of my own vocal means the « normal » pronunciation ([bɛʁgsɔ̃]) is almost impossible if I do not slow down the elocution ; only when I opt for a leisurely articulation can I obtain a clear enough [g] ; otherwise, given a usual rate of elocution, this phonetic consonnant invariably finds itself diverted into [k] ; in this environement the voiced velar plosive g is devoiced, thus producing the voiceless velar plosive k, the lenis consonant passing into its fortis counterpart.

Wikepedia

Later studies have shown that articulatory strength is not completely irrelevant. The articulators in the mouth can move with a greater velocity and/or with higher electromyographic activation levels of the relevant articulatory muscles with fortis consonants than with lenis ones.

There seems to be reciprocity: greater speed tends to impose fortis consonants.

II

The fact that final "on" can be [ɔn] in French instead of the usual [ɔ̃] is due to the influence of the foreign pronunciation of this combination, in particular English. For instance the pronunciation is [edisɔn] pour "Edison" and never [edisɔ̃].

The article is not wrong, Bergson has always been pronounced /bɛʁksɔn/, as you can hear in this archive audio document from 1936 (https://www.youtube.com/watch?v=S1IBlecNs8U) where the philosopher’s name is pronounced at 0’14” and again at 0’23”.

It is expressly recommended as /bɛʁksɔn/ in Philippe Martinon’s pronunciation guide, Comment on prononce le français, from 1913 on p.148, note 4 (https://archive.org/details/commentonprononc00martuoft/page/148).

It might be due to Bergson’s English connection through his mother. He only became a French citizen when he came of age, although he could have chosen English citizenship (cf. https://plato.stanford.edu/entries/bergson/#1).

/g/ becomes /k/ because of assimilation with /s/. /s/ is devoiced as opposed to voiced /z/ and, in connected speech, /g/ loses voicing and ends up being pronounced /k/ giving /bɛʁksɔn/. This is a widespread phenomenon in French, ça va de soi, for example, can be realized as /savadəswa/ where /d/ is not immediately followed by /s/ and is not devoiced and /savatswa/ where /d/ assimilates to /s/ and loses voicing.

In fact the pronunciation is phonemically /bɛʁg.sɔn/.

The final is not nasalized because it’s considered a foreign name. Foreign names are generally read more or less letter by letter, although there is considerable variation depending on the name and on the speaker. Until the 18th century (I think), foreign names used to be transcribed into French, then pronounced using the normal rules of French. These days, for names that were originally in Latin script, the spelling is not adapted but the name is pronounced differently. For example Lisbonne, Francfort, Rome, Milan, Bombay, Pékin are pronounced the way they’re spelled, which is different from the original name. Names such as Dublin and Berlin are spelled like the native name, but pronounced as if they were French names, with a nasal vowel [-ɛ̃]. More recent imports are pronounced in a way that approximates the original pronunciation slightly more closely: Washington [wa.ʃiŋ.tɔn] or [wa.ʃin.tɔn], Medellín [me.de.lin] or [me.dɛ.lin], Amman [a.man]…

With people’s names, an added factor is that it is to some extent a choice of the people themselves. It can happen that different members of the family pronounce the name differently. For all I know, Henri Bergson might have had French cousins who pronounced their name [bɛʁgsɔ̃] or who spelled it Bergsonne.

As for the g, I don’t know how Bergson himself pronounced it, but both my philosophy teacher and the 1936 interviewer found by petitrien pronounce something close to /bɛʁg.sɔn/. The /g/ sound in French is nominally a voiced velar stop, while /k/ is a voiceless velar stop. However, the transition between the voiced /g/ and the voiceless /s/ is difficult, so its phonetic realization is somewhat devoiced. (This is the reason why x is pronounced either /ks/ or /gz/: either both consonants are voiced or both are voiceless.) The /g/ in Bergson is still perceived as a /g/, though, not as a /k/, even though it is phonetically somewhere in the middle.