Pushbullet
Text from your computer
If you enabled SMS sync in our Android app, you can text from your computer via Pushbullet.
Texting from your computer is easy, here's how:
Click SMS in the left bar.
And there was no "SMS" in the left bar.
Maybe it works better with Android.
-Steven Avery
Pushbullet does more than that. I have an extension, and I can do exactly what you're doing.

This is what I see in Chrome (Brave) with the extension. The People tab is all of my pushbullet native contacts (you can communicate just using it), the Devices tab (the current tab) shows what device I have linked, and you can see a couple of the links that I've sent to my phone. The SMS is all of my SMS from my phone, and I can send SMS with it and receive it. The notifications tab has all of the notifications that I see on my phone (I can't react with them, i.e. when it gives me choices in the notification, I can't select one- that's a paid feature).
One of the top reviews from the iOS store:
Been using Pushbullet for years. Essential for desktop to iOS
I love this app! Whether it's to save things for later,send it to the Pushbullet app so I can text it to some of my friends as sometimes I'm on my custom built PC and I can't airdrop like I would on my MacBook. The features are there and they're so many more functions besides what iOS can do alone like remote files, multiple device management, sending it to people within Pushbullet. It's great, It's wonderful, It's been working for years and I totally recommend it.
Something I want: Can we get an iMessage app so I don't have to tab to the Pushbullet app copy a link or hit the share button go back to the Messages app and vice versa. Being able to just grab right from Pushbullet in a small little iMessage feed and paste it right into a conversation that's already open would be amazing and super fast.
So apparently the same works with iOS.