I've still managed to lose a few fish over time to sickness, which on reflection will have been down to insufficient water changes and/or cleaning out filters
This would take a lot of info to possibly sort out, and I am not much on disease issues having had so very few in my 30 years keeping fish. There is a lot involved. We can move on.
So presumably the plants are not enough to 'eat' all the ammonia that's produced.
Possibly, but you do have some faster growing plants (the stem plants), and these are your better consumers of ammonia/ammonium than say ferns, mosses, and Anubias. There is some pennywort on the upper fight in the photo, I would develop this into a floating plant. Untangle the stems and lay them across the surface. This stem plant makes a very attractive floater. The leaves will all turn to lie flat, and the roots will dangle into the water. I had good results with this plant. Some other good floaters are Water Sprite, Frogbit and Water Lettuce. Water Sprite is my overall favourite, in my view every tank should have this floating, it is an incredible water purifier--often called an ammonia sink because it can assimilate so much ammonia/ammonium.
So one is still relying on a mix of plants and other filtration material, plus water changes to deal with the nitrates being produced as a result of the bacteria's work. Is this correct?
Yes and no. The plants' capabilities (as I just discussed above) can result in most of the ammonia being immediately taken up. And this is non-stop, day or night. Some ammonia will (so they say) still feed ammonia-oxidator nitrifying bacteria/archaea, and what does will naturally cause nitrite and then other bacteria resulting in nitrate. But I have never been able to measure ammonia or nitrite in any fish tank--and some of them were quite heavily stocked I thought. Nitrates always measured in the 0 to 5 ppm range using the API liquid test, which may have been 0 or 1 or 2 or...5 ppm for all I know. But it never went above that first colour range, except once in the 90g and that is another story. You want to see zero ammonia and nitrite obviously, but you also want nitrate at zero or as close as possible, permanently.
You do not have a heavy stocking here, so I would never expect to see these three if all is well. Assuming nitrate is zero in your source water, the plants plus significant water changes should keep nitrate as low as possible.
Water changes should be regular (once a week) and substantial (50%, or more--I did 60-70% for years. Provided the parameters--which here refers to GH, pH and temperature--of tap water and tank water are basically the same, you cannot do harm with water changes. No fish in its habitat lives in exactly the same water from one respiration to the next; we cannot hope to match that, but it does exemplify the aim.