I will be comparing advantages and disadvantages of separable verbs in German as reasons for their continued existence.
Pro: Part of a syntactical construction
It is possible to analyse separable prefixes not as prefixes, but words on their own. This is also the most probable source of those words, see the question's comments. And grammar changes slowly, so that may be the main reason we still have them.
In this view, they behave exactly as they are expected, so having them is actually less cognitive overhead than one would think. Compare
Er (schlägt/tritt/macht) die Tür (auf/zu/kaputt).
All combinations are possible, and the first word does not influence the meaning of the second word and vice versa. These combinations are in a sense "transparent" and very productive. The position can also be filled with longer phrases that are unsuitable as prefixes like
Er schlägt die Tür klitzeklein / in tausend Stücke.
An "advantage" of separable verbs is thus that it is part of a more versatile grammatical construction, which constantly produces new separable verbs.
Another source are noun-verb-combinations, like "Bahn fahren", "Leid tun", which behave a bit like separable verbs. Some of those constructions get inseparable, like "staubsaugen".
Con: Hinders comprehension
The big disadvantage is that especially if the combination has a vastly different meaning than its constituents, the entire phrase becomes harder to understand if the hearer has to "reparse" upon hearing the particle.
In German, this is somehow mitigated by the fact that separable verbs often have different valencies than their base verbs, like "sehen-transitive" and "umsehen-reflexive". But having to wait until the end of the sentence to get the full predicate is common in German, so this might not be such a big disadvantage in the eyes of German speakers.
Chinese, for example, also has something that could be considered separable verbs (verb-verb and verb-noun combinations), but severely restricts what can be inserted into the gap because the huge amount of homonyms makes it hard to understand which verb is meant without the complement.
Pro: Close to the object
Most of the the separable prefixes are describing positions of the object and as such, it makes sense to put them near the object. In the V2 case, the prefix would than be close to the subject and not the object.
Consistency for each prefix
Maybe to keep the cognitive load for keeping those two types of prefixes apart, in German every prefix is either separable or not and this does not depend on the verb. It also almost coincides with whether the prefix is stressed. (except for e.g. miss-) Keeping them apart is thus also kept simple.
But why did we start to have prepositions as particles? I don't know. It seems like Germans perceived a verb with a preposition as something different than without and then decided to keep the preposition even if there is no noun anymore.