Though from what I've read in reviews the Osmo doesn't have too good of a battery life or audio, so not necessarily a good choice as a dedicated camera for vlogging. But the mirrorless camera's are what I'd go for too. Below is my progression as a daily vlogger:
1. Olympus PEN E-PL6 kit with lens, EUR 300. (I think the newer E-PL8 model is now the same price)
2. Sandisk 32GB Extreme memory card, EUR 29
3. Gorillapod Hybrid, EUR 50
4. 20mm f1.7 lens, EUR 250
5. Olympus OM-D E-M5 Mark II body, EUR 1000 with an extra battery and a memory card
6. Rode VideoMicro, EUR 50
7. 10mm f2.4 lens, EUR 350
8. Gorillapod SLR Zoom, EUR 70
8. DJI Phantom 3 Advanced, EUR 799
9. Another Rode VideoMicro since I broke the previous one, EUR 50
10. Rode VideoMic Stereo Pro since I broke the VideoMicro again, EUR 129
11. 45-150mm f3.5-f5.6 lens, EUR 99
12. FalconEyes mobile light, EUR 39
Even if you don't go Olympus but go for a mirrorless system, you can use the Gorillapod Hybrid to improve ergonomics a lot. You can use it as a tripod on the go, as a selfie stick, support it against your body if shooting other people etc. As long as the camera + lens combo weighs less than 1kg you're good to go. Also, with this setup, you could deviate at item 5. and just get an external recorder to improve audio. That was the main problem with the PEN. No 3.5mm plug for the mic and the onboard mic is cell phone quality.
Turned out to be a pretty long post, hopefully you'll get something out of this. Also, most prices were deals and I live in an expensive country so your prices could be different.