Why did English lose "-an" endings?

These are two different things, though with the same cause (simplification and laziness). Firstly the infinitiv of most regular verbs had a -en suffix or ending. For example in modern German - tanzen, essen, schwimmen. Over time the infinitives have shortened for ease of use and lost the suffix so only the root remains, ess, tanz, schwimm, for example.

The article you cite relates to the simplification from a declined language to one without significant declension by firstly removing dative and accusative cases and secondly standardisation away from three word genders to just one.

Old English unstressed a regularly turned into Middle English e

The change of -an to -en is because of vowel reduction. This was a regular sound change between Old English and Middle English that turned many Old English unstressed vowels into Middle English "e". The pronunciation of Middle English unstressed "e" is often reconstructed as [ə]. In later stages of Middle English, and in modern English, such reduced vowels came to be lost entirely when in word-final position, and sometimes when followed by a word-final consonant.

This same sound change turned the Old English plural noun ending -as into the Middle English plural ending -es.

Middle English final n in unstressed syllables was often but not always lost; the reasons are difficult to determine

The loss of n at the end of word-final unstressed syllables is more complicated. Because final n survived in some words and some grammatical forms (such as oxen, driven) but not in others, the loss of n in this context does not seem to have been a completed regular sound change, at least not in all dialects of Middle English. I found a paper on JSTOR about the loss of final n; however, its analysis is focused on words other than verbs: "Loss of Final n in Inflectional Syllables of Middle English," Samuel Moore, Language, Vol. 3, No. 4 (Dec., 1927).

Moore says that some Middle English texts, such as Owl and the Nightingale, show almost complete loss of final n in unstressed syllables; some texts, such as Cotton Nero A 14 Ancren Riwle, show loss in certain noun forms and in adjective forms, but not in verb forms; and other texts, such as Chaucer, show loss of n in singular noun forms and in all adjective forms, no loss in the plural noun marker -en, and variable loss in verb forms (232-233).

Moore makes the point that the distribution of n based on grammatical categories in these texts indicates that we are dealing with more than just a purely phonetic phenomenon (if all that was going on was an optional phonetic loss of n, we'd expect loss of /n/ in each word to be based on the phonetic surroundings in each particular environment, not on the part of speech or grammatical number of the word) (233-234).

Unfortunately, Moore's paper sets aside the loss of final n in verb forms as a problem requiring further study because this loss seems to have occurred later than the loss of final n in words of other grammatical categories (page 255-256).

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The following source also mentions variation in the use of final n on verbs in Chaucer: Introduction to the History of English (HoE): Morphological change: Verbs, Hans Platzer, University of Vienna.

At this point, I don't know enough to tell you why n was eventually completely lost in Modern English as an infinitive marker.