Oh yes it does.
I don't see how. They'd have to retcon the first 2 seasons of VOY in that case - and since Prodigy is a sort of 'continuation' of VOY (or closely connected to it), then I doubt they would retcon all Kazon worked for the Diviner in 2360-ies... and that just doesn't track with what we know of the Kazon as a whole (at least those on the other side of the galaxy).
I would imagine that this particular Kazon had others with him when he found a TW conduit to the DQ... closer to the BQ/DQ border (in the galactic core) where Tars Lamora is located - so that ship and crew seemingly began operating in that part of the galaxy instead and just ended up being hired mostly by the Diviner.
All in all it was a good episode.
A little background info on the crew of the Protostar, and its surprising how Janeway managed to (nicely) deduce the kids have simply found themselves in over their heads... and she understands better their motivations (especially after being slaves, sold or bartered by other individuals)... but what she doesn't understand is why they refuse hails (obviously because the living construct could infect the Dauntless and the rest of SF, but Janeway doesn't know this yet).
And dangit... she just HAD to enter at an inopportune moment and was knocked uncoscious (at least she wasn't killed).
Things are certainly getting interesting.
Chakotay and his crew are seemingly stuck in the future on Solum still - will be fascinating to see how/if they manage to get back - maybe this will end up being the driving point of the second season once the immediate situation in the late 24th century is resolved, but with the possibility of time travel and how it could lead to unpredictable outcomes... maybe SF in the present will have standing orders to send a rescue party to Solum when the time comes - so Chakotay and his crew might end up remaining in the future for all we know - and I was at least somewaht right in deducing how the Protostar managed to end up where it did. Plus, the dialogue made it seem like the 'Order' sent people intentionally back in time with tech they had at their disposal, but it turns out they used an unstable wormhole which brough the Protostar to them in the first place... so it stands to reason they'd be deposited in random points in time searching for the ship.
It seems the Protostar ended up traveling through a wormhole by accident (but we don't know if the wormhole itself was a natural phenomenon or if it was something generated by the Proto drive/core).
If it was a consequence of using the proto core, it would explain why it would be considered 'unreliable' (however, this doesn't track entirely because even misaligned Warp engines can create Wormholes as evidenced by TMP and Warp wasn't abandoned as a result). Thus far, the Protostar jumping twice with the teens onboard did NOT result in time travel.
I'm inclined to think (for now) the wormhole was a freak (natural) occurrence and had nothing to do with the Proto drive.