you could wait until the new timber ages to match the side fence or paint the old and new fence a matching dark grey

I’d stain the fence dark then maybe add a couple of decorative screen panel. Look at the range at Chippys outdoors or Bunnings.

Love above suggestions, paint fence a dark colour, this highlights planting, add screens , you can have them backlit for nighttime use, or a textured wall panel would look gorgeous

Personally, there is nothing wrong with that fence. It's a nice paling fence and 6 months in the weather will have it looking identical to the rest of your fence.

With regard to not being able to grow much under those trees. Lomandra is a large grassy tuft with serious landscaping cred and will grow directly under eucalypts, it's where they grow out in the wild. They are also tolerant to full-sun to almost full-shade in terms of aspect. Black boys (grass tree's) will also grow happily there and cover the fence with a very nice look. Throw some large bush rock in there and plant out some nice little rockery plants (clivia will do well with that shade) and you've got yourself a nice little backdrop. But yes if you have large tree's it's not ideal to try and plant directly into their roots. They will feed all the nutrients out of the soil. Planters or raised bed is best.

Which side of your yard faces north, south, east, west? That will help a lot.

My own yard gets full sun only in the early morning and parts of it are full shade from midday onwards. I am growing a Japanese styled garden there as it gets protection from afternoon sun but has enough direct sun in the morning for items like azalea's, maples and acacia cognata (weeping wattle). The grass is also a moss, rather than lawn.

Work with the landscape as it is rather than trying to fight it. There are plenty of plants that are very happy with minimal direct sun and you can do a lot with a partial shade yard. You're just not going to grow a veggie patch, or fruit trees in that section of the yard, it looks like you have full sun aspect along the other fence which would be better for that kind of thing.