A few things to consider:
1) A lot of those cheaper lenses have very flush rear elements (the glass at the back of the lens is flat with the back of the lens); whilst many teleconverters have a protruding front element. This means that many won't even physically fit the back of such a lens as the front element pushes too far forward to allow mounting or it can be too wide to fit into the gap of the lens when there is room.
Zoom lenses can also move elements and the rear element might move which could mean its fine at some focal lengths and smashes into the teleconverter at others.
So you'd want to research which TCs will actually mount
2) As noted the effective aperture of the lens will reduce and this will have a few impacts
a) On that body if the camera detects the TC, it won't allow you to use the AF at the long end (if at all). This is because once the combined aperture gets too small the camera shuts off the AF as too little light will get to the sensors to provide reliable AF.
b) Due to reduced light (even when wide open) the AF (if present) will be slower to lock on and might well hunt a lot more.
c) Your range of apertures and thus shutter speeds is affected. You'd find you'll have to raise the ISO up to get enough shutter speed even when wide open.
3) Image quality reduces. TC's magnify what a lens shows and that means enlarging things. Now in general a TC will give you better resolution than cropping and resizing will to get the same display. However for most people a 2TC takes a lot of image quality away and they find it creates unpleasing results that are just too soft/lacking in detail to make them worthwhile.
On a cheaper zoom like your 70-300mm you don't have a lot of room to play with image quality before it will make a very noticeable hit.
Honestly I'd not advise it on that lens. A 1.4TC might be more practical (if you can find one that will mount) as their impact on all regions is far less; but even so its still going to be a noticeable difference on a lens like that.
I'd start saving pennies and keeping an eye on secondhand traders. There are some very fine EF and EFS mount lenses out there and you might well find something like an f5.6 400mm L that will give you the reach you want and be far superior on all fronts over the TC. Yes it requires saving for longer, but the end result is something that you are much more likely to get pleasing results with that make you want to keep using it; rather than a 2XTC that might well just prove impractical