Thanks so much for your replies!
After a couple of days, my own troubleshooting, and a bit of help from Convertkit, I managed to embed the HTML version on my 7.1 site without having to use Zapier. I'll detail all the steps below:
Then, I went to my Squarespace 7.1 site and added a 'code block'. By default it's on HTML. I left the 'Hello World' stuff there just as a placeholder, and pasted in the embed code. It's pretty long. Then I hit 'Apply'. That made the form appear, but the words 'Script Disabled' were still above it.
What I did next was I used Ctrl F to find all the instances of 'https' in my form code. You have to remove the 's'. I'm not sure why that fixes it, but it does. So I found 3 cases of it in the form I used and changed all of the 'https' to 'http' in the code block. Then I hit 'Apply' > 'Save' on the site, and reloaded it. And it works fine now.
I also tested it and it ended up sending my emails perfectly. I have only tested it with one address so time will tell if it continues to work but I'll post here again if I have issues.
You can definitely still use Zapier but I'm ok using the HTML code. With Zapier you only get 5 zaps to start I think and there is a 15 min delay with each sign up on the lowest plan. It can get very expensive when you are just starting out as a business owner (like me!) so I am trying to keep things low!
Let me know if you have any questions, always happy to help if I can!