What we've gelost — why doesn't English use the prefix "ge-"?

Solution 1:

In short:

• In Proto-Germanic, the prefix was *ga-;
• In Old English, it was ġe- (pronounced /je/, /jə/);
• In Middle English, it was y-, i-, or ȝe- (pronounced /ɪ/);
• In Modern English, it survives in a handful of words as i-, a-, or y- (see below).

The Wiktionary page for y- has these usage notes:

This prefix represents a common Germanic perfective prefix which was used to form past participles. Already by the Old English period such participles could be used with or without it, and as it passed into Middle English forms y-, i-, and ȝe-, it became less productive. The prefix was later adopted as a conscious archaism by some writers such as Edmund Spenser, who prepended it to existing past participles.

Etymonline has this to say about y-:

perfective prefix, in y-clept, etc.; a deliberate archaism, introduced by Spenser and his imitators, representing an authentic M.E. prefix, from O.E. ge-, originally meaning "with, together" but later a completive or perfective element, from P.Gmc. *ga-. It is still living in German and Dutch ge-, and survives, disguised, in some English words (e.g. alike, aware, handiwork).

Finally, the Merriam-Webster has this discussion of yclept:

"Clepe" itself is a word that is considered archaic and nearly obsolete, but its past participle "yclept" (pronounced ih-KLEPT) continues to be used, albeit rarely. In Old English, the prefix "ge-" denoted the completion or result of an action; in Middle English, the prefix shifted to "y-" and appeared in words such as "ybaptised" and "yoccupied." Eventually, all the "y-" words except "yclept" fell into disuse. One reason that "yclept" persists may be that it provides a touch of playfulness that appeals to some writers. Another may be that although "yclept" is an unfamiliar term to most people, its meaning can usually be inferred from context. Whatever the reason, "yclept" continues to turn up occasionally in current publications despite its strange and antiquated look.

Emphasis mine in all cases.

And yes, I realize that I haven't addressed the why part of your question.

Solution 2:

Like German, Old English did use ge- as a prefix to mark past participles. As it moved into Middle English, this evolved into y- (also i- or ȝe-), and as with many forms of inflection became non-productive and mostly disappeared by the time modern English rolled around.

Wikitionary lists yclept as a holdover, though that in itself isn't terribly common. It does illustrate the point well though moving from geclypod in OE, to ycleped in ME and now to yclept.

Solution 3:

I can only tell you that Old English had the ge- form. For example, the inscription on the Ælfred the Great Jewel says "ÆLFRED MEC HEHT GEWYRCAN". That translates to "Alfred had me made [crafted]." And gewyrcan would have been pronounced "yewirkahn", roughly speaking.

That said, John McWhorter cites the loss of these prefixes (along with be- and for-) as part of what the Vikings did to English. They simplified many forms and caused many markers to be dropped. (See Our Magnificent Bastard Tongue.)