Just to be clear - G-sync / Freesync are not at all required to attain higher refresh rates. As far as I can tell, the important thing for you here is the higher refresh rate, not necessarily the variable refresh rate (although of course you will want to make sure the tech is available if upgrading).
Also, FYI - your issue with 60Hz is quite possibly the cable you are using (depending on the HDMI output of your current laptop, which is probably an older version, and will in that case not handle that resolution) - you need to make sure you are using an HDMI 2 > to test this, but if you are upgrading, not a huge issue.
The crux of it is this - you should be able to attain 1440p @ 144Hz via HDMI (will not do 165). Do not waste your time trying to get variable refresh rate tech via HDMI - it will - under specific conditions - work, but really not worth the hassle. That said, it does look like it's possible, but here are a lot of people struggling with this issue (with your particular monitor); looks like there may have been an update that allows 144Hz @ 1440, but it is not 100% clear that this will also support variable refresh rates.
Make sure you are using an HDMI 2.0 cable for whatever you do (obviously).
Also, there are adaptors that convert USB-C output to Display Port, however, I have no idea if VRR will work if you do this.
Basically, if you are upgrading your laptop, try to find one that has a DP output - this will solve all of your issues.