You don't need to chop them down if branches are positioned where you want them. I think you should certainly remove branches more than half the diameter of the trunk at point of their attachment- especially if there aren't 3 branches around that size, spreading in different directions. I usually remove branches more than a third. Oversized branches can choke out the leader and imbalance the shape of the tree. Sometimes heading back main (scaffold) branches helps to generate secondary branching.

So I was searching for posts on Sycamores, and I stumbled on this post from me. I thought I would post an update. Here is a picture of the same Sycamore just 2 years & 3 months later. It is at least 25 feet - maybe 30 feet tall now, and doing well (have some trouble with powdery mildew, but I'll live with it). I do love the Sycamore tree. The Dawn Redwood is in a bit too much shade probably. It's probably 20 feet tall now and doing fine - if not particularly filled out with leaves. I'm looking forward to when it breaks through the canopy and really shoots up and flourishes.

I like the lattice. That's new too isn't it? Don't recall it from your last photo. Do you plan for clematis for the lattice area? It will grow happily entwined up into the stair railings so if you think the railings will need painting/staining in the next few years, might want to do it in advance of planting. That way you don't have to chop back vines for several years. The foliage will help keep the sun effects off the wood so the paint/stain lasts longer. Clematis has a certain bloom time and span but you can put annual vines like cardinal climber to bring color to an already bloomed perennial vine.

with a new planting.. i would not prune for at least two seasons ... the first to let the roots settle in and get established.... the second to grow free .. to work towards the goal i wanted to accomplish not know the size you have now.. nor your goal.. nor a pic.. its hard to give any real advice ... pruning is not really requisite.. for the hell of it ... lol ... what is your goal?? ken ps: as one examply .... usually privet is sheared into a box-like form ... but one might prune it into a topiary design ... so a good start .. would be looking up the terms.. and using the one that more closely aligns with your future goal ... a good start at the link.. flip to the ALL side.. and look at the first link ... after reviewing something like that.. you can narrow down your goal.. then we can offer much more precise info ... https://www.google.com/search?q=privet+pruning+vs+topiary&source=lnms&tbm=isch&sa=X&ved=0ahUKEwimqYnjhLHKAhWBKCYKHdnaD4cQ_AUICCgC&biw=967&bih=745&dpr=0.9