For gardeners to comment you need to tell us where you garden. If you are in the US, what zone?

In Illinois burning bush is considered invasive. I just had a whole hedge of them removed for that reason.

Ooops! Zone 6 or 7, north of Seattle and near the Salish Sea. The coldest I know of was -5 fahrenheit with wind chill too. We're getting stronger and stronger wind storms. I don't want to put in stuff that would blow down.

I don't believe burning bush is invasive here but I'll check. They sell them at local garden centers. Mine has not spread in my yard. I pulled out butterfly bushes, yellow flag iris, and Autumn Traveler's Joy clematis when I learned they were invasive. Vinca is only in pots and hasn't escaped in 10 years.

Let me find some photos:

The first photo shows the old wood fence (over 20 years old), and a bit of metal fence we could replace it with. The goal is dog-containment for my neighbor, and privacy for both of us, but not "Go Away And Leave Me Alone" kind of privacy, just "you don't notice if you're not trying to look" kind of privacy. Yes, that's Moss Hart, a moss topiary with real antlers. Yes, he's life-size. Yes, he's hard to maintain because the squirrels adore him. They burrow and tear off his moss running around him as a jungle-gym, and also eat his antlers. I love him anyway, and enjoy watching the squirrels duke it out with the blue jays over him. This photo shows the other side of the yard from where the fence blew down. But it shows the kind of fence that blew down. It's time to replace all the wood fence. I've been replacing all the lawn with sword ferns, deer ferns, and other local evergreens. I need something taller for privacy. There's a big old pine tree just off camera to the left.

The next two photos are the fence out front when it was brand new, before anything had a chance to grow around it. It's restored salvage from the 1800s from New York brownstones. I've got another 40 feet of it we could use to replace the blown-down fence, but it doesn't give privacy, except with added greenery. I wanted a fence out front that flirted: "Don't you wish you could get invited in?" I invite people in as often as possible. If you live near Belliingham, ask for a tour. I'll pump you for advice!

Here's close-up. No two panels are exactly alike, which I just love, though they are all very similar. We had the sections out front restored. The best we'll be able to do with the 40 feet that remains will be a few coats of black rustoleum and get it bolted onto powder-coated 2x2 metal posts with some kind of ornamental caps on them. The next owners can pay for their own restoration. The next owners will be after we're dead and gone.

Here's the front of our house so you can see the style. You can barely see the fence in this one. I let it get kinda overgrown in this shot. The house is a 1905 Folk Victorian.

An evergreen hedge is much preferable to something that loses leaves i the fall Where you are conicla cedars are always a good choice and your euonymus would work but they tend to spread out more that grow tall. Gorgeous yard BTW looks like a lot of work so maybe keep the hedge something easy to care for. I live in zone5-6 in Creston B.C. and I think the zone afford so much choice in plant material it is almost overwhelming. We moved here from Calgary so I went crazy in my yard. My SIL has holly as hedge in her yard it is really quite beautiful she is in vancouver.

Is the old metal fencing that remains enough to contain the dogs? You could interleave simpler metal sections if you need more.

I did a living fence around the front yard. I am zone 5. I don't like the formal look and used various bushes and tall grass. The grass is tied up in the fall and adds much needed texture to the yard in the winter months.