Yep, I've got about $200 invested into my "gravel bike", and it works better than stock, and I'm proud of that. You don't have to spend a thousand dollars to have a functional bicycle.
As for moving the COG forward, sure it does (a little bit). But you can move your COG backward by moving your butt back off the seat and straightening your arms out. Just like you do on any bicycle any time you go down a steep hill. Even with drop bars you can go down stairs or steep hills without going over the bars unless you grab a handful of front brake while doing so. Or if you're like me and you still haven't perfected the art of the endo
There are all kinds of videos on youtube of guys doing amazing things on road bikes; drop bars will not make your bicycle totally incapable. "Road bike party"
If you're not happy with flat bars, and you think drop bars might be more comfortable for long rides, you're not alone. You can find conversion videos on youtube, but it looks like your bike might already have a 31.8 stem which would make a drop conversion super easy.

Edit: And, if you use the same size stem, it doesn't move your COG fwd at all if you ride in the upright position. Drop bars are just versatile like that.