Since the Panasonic LUMIX DC-FZ80 is out of stock everywhere, I decided on getting a Canon PowerShot SX60 HS instead.
It has all of the features I could ask for and then some.

Can anyone that has used this camera before tell me their opinion on what it is like to use?

Do yourself a favor, get a large sensor camera, DSLR or mirrorless. Some used ones, but still working well, cost next to nothing. Small sensors, like 1/2.3 here are helpless in dim light, whatever the manufacturers say. Just because of plain physics, they can not capture enough photons in a small (absolute) lens aperture. Consumer toys like you are mentioned are grossly overloaded with features to capture the attention, but perform poorly on their main functionality. You are generally getting a performance of mobile phone camera in a *much* bulky case.

Yeah, the larger sensor cameras really excel at night, especially with paired with a wide aperture lens. I'd reckon for the same price as a PowerShot SX60 HS you can get a used APS-C DSLR with the kit lens and a good prime lens for night shots or a telephoto lens for closeups.

If you still prefer the point and shoot style cameras, you can get the prosumer ones with a 1" sensor which performs a lot better over the consumer cameras.

If I want to do low light photos, then yes I’ll use a proper DSLR (I am looking into a Nikon D3200) But the issue is I can’t zoom in very far with the lenses that those have.

And besides, for most of my regular photography I will end up using it with a proper flashgun that turns night into day anyways, so low light performance doesn’t really matter for what I’m going to be using this camera for most of the time.
So in the end, will still end up needing to carry around two cameras that each do their own job.
Do you still think it will work for my application?

Flash works well for wide and normal focal length, somewhat for moderate tele. No way for it to light up the object at extreme tele (at about 1300mm equivalent for this camera). You can calculate (using solid angles) what proportion of your flash energy will reach the object, and what will be lost elsewhere around.

I looked up (out of curiosity) some sample photos of this Canon. Image quality at long end is just awful, as may be expected. Not sharp and full of noise, though aggressively processed by noise reduction algorithm giving a watercolor look. Phew... Image at wide end may be better, but still not sharp and noisy even at ISO100. And camera dynamic range is rather poor giving white sky and black shadows. I won't touch this, honestly. Competing Panasonic (70's series) looks notably better, but still very poor at long end.

In the real photography application, you rarely need a focal length longer than about 300mm equivalent. This is covered by 200-300mm lenses for FX and APS DSLR/mirrorless and 10x superzooms on compacts, which may be still stay within a reason for image quality.   

I think maybe I’ll get the D3200 and just keep the Panasonic DMC-FZ35 that I have now. 18x is plenty of zoom.

This is a much more mature choice, of course. If shopping for Nikon bodies, I would splurge a bit for something like D5200 or later. 5 series will have *slightly* better image quality, but notably less restricted handling and user interface. Also do not forget that some older pro and semi-pro FX bodies, like D3, D3s, D700, D750, D800 and so are starting to get their price into a sensible range. Shopping for used cameras is tricky of course, beware of high shutter actuations count, but using these will be a whole another  world of photography, and you will be able to mount literally millions of classic Nikon F lenses floating around without any limitations.