Besides vor-, which is an amalgam of at least five different pre-forms, there was also a root *pers-, which one might expect to have contributed here. I mean, I have no idea, but am concerned with the problem that a rather frequent verb such as schlagen "to beat, win,..." is known to have no certain etymology. You knew, right, that's why you are asking? This would leave the question what -(h)lagen would be.
klagen, as though "to claim". Although this is rather negatively connotated with suffering and complaint because of the legal field, which might be well old in this respect.
- Ankläger is possibly the original etymon to an-klagen, if An- rather belongs with ander- (that still meant "second" with Luther, similar for En. other), the same way that we call Zeugen (unbeteiligte) Dritte. I am not sure if there are cognates with the preposition for this particular motion.
- Problem might offer semantic parallels. Its Gk. stem balle- "throw" should belong with Qual "torture" according to official sources, both words from a Proto-Indo-European root assuming a sense "to punish". A Problem or Vor-wurf is not synonym with Vorschlag, so it explains very little, but the coincidence is still funny though. (PS: as vor Vorwurf, despite the apparent stem, consider a semantic tangent to PIE *wer- which reflects in words for oaths and speech; well, same for *b.lH- and ballein)
legen, lage, Vorlage. (PS: s.a. überlegen, Überlegung) I wont grasp the full extent of the jam, but this works various ways. First, there can be no argument that Vor-lage, a noun that exists, would preclude a doublette, since it could be a more recent formation. Although I have to admit between this and that, it doesn't matter much to what end the s has sprouted. As English slay would suggest, there never was a c /k/.
- in Football, Vorlage is an assist, a pass from the left field for somebody else to finish.
- in Design, it is a template, a passpartout from the left field for somebody to finish, a sample, a form, cp. project, projectile
- in accademia, it is a standard work, a right of passage from the left field for somebody else to finish, I guess. I mean, the participle vorgelegt is met very often in book reviews, and I'm not sure if it always used with such a specific meaning, or rather colloquially to mean "submitted" (in which case it would mean the reviewer didn't have to pay for it)
- in general, the preposition can modify the verb in the spatial as well as the temporal domain, although we have other verbs for similar senses, like lagern, vorverlagern (compare Verschlag, too).
- conversely, the verb nachlegen is similar to nachschlagen. In general, both mean to repeat, do the same, although the specific meanings differ slightly. The basic semantics seem to pertain to ovens, for example--put in more fire, get more heat, so nachlegen in general means to double down, reload, retry, attempt.
- In research, nachschlagen means, well, re-search in particular, and it seems to riff on the idea of search by slamming books open, up and down, and finally shut. Here too we can refer to legen as of a lever, einen Hebel umlegen, as of a book page (theoretically). At this point a fossilized infix morpheme -s- of a gone casus as in -wards would be imaginable. Jedenfalls ...
- on the other hand besides the literal sense it may also mean to develop in the same direction[''can this be sourced?'']. Logically, the response to a Vorschlag should be a Nachschlag, if vor and nach are opposites. They are not exact opposites, but both constructions exist and should show parallels in similar domains. This abstraction may be figurative from the sense "to throw", cp. Schlagball as a sport, see above problem. This is also the most plausible explanation for Einschlag "characteristic", einschlägig "typical" (PS: notwithstanding what type of verb the original context would have required).
- to stick with legen: we might note that earliest attestations with Grimm trace einschlagen to a sense of staking out a pillar so as to start a mine, set up shop, which is reasonably similar to anlegen, also compare Anlage "attachment* and einschlagen "include, envelope" (e.g. figuratively in a testament, cf. Grimm). There's more, I'm sure.
lean, leaning, inclination (from *'kel-, *' kley-) has cognates that overlap in meaning with some of the above. The velar (*'k) is typically lost to aspiration (cp. e.g. house*, Lat. casa; (h)raven, crow) but persists in affrication with any arbitray leading s-, cp. Schlüssel, Schluss, Lat. clavis (claviatur = key-board), cludo, En. close (from *(s)kleHw-), further lock, Ger. Losung "keyword, passphrase; motto" (probably belongs here, too). I have no idea how this should connect to vorschlagen, other than to stand out, step forward. (PS: An exact opposite meaning might be with secret keeping, occult, see Verschlag, unterschlagen).
It could of course be from the very schlagen similar to Two above, potentially hiding a sibilant in the vor-syllable. English cognates seem to be confined to violent meanings, slay and slaughter and a tentative root is proposed based on this which has little external evidence, while slack (viz. flat, cp. layer) might or might not belong here.
You know, I really doubt problem came from throw (go balistic), and am rather looking at blame, blemish, or so, which might be however later derivatives.
- Another note deserves, probe, prove, like try, trial (the later two surprisingly not as closely rated as I had thunk). This is directly from pro-, so it stands to reason that vor- was likewise an ablauting preverb and in this sense the head of the phrase. This leaves the tail as a possibly morphosyntactic extension akin to -like, -ly. In this view it is noticable that Kluge/Seebold make reserved mention of "Menschenschlag" or "vom gleichen Schlag" (of the same ilk, flok).
Finally, checking the initial idea, we find that PIE *pers- > *pŕ̥sos > Old Norse fors, foss "waterfall" is not really compatible, but there are a number of other *p-r- roots with *s in the derivational morphology such as for *preḱ- + *-sḱéti > *pr̥sḱétito > Ger. forschen cp. fragen, Lat. prex, proco (and let me say it should be curious if procuro did not belong here as well).
Disclaimer: All *roots have been copied from en.wiktionary--all mistakes are, as per usual, mine.