I think it'll eat 7000 connection objects, maybe threads, but they'll all be on port 80 or 443? So if you can keep the overhead of each connection down, presumably easy because you don't need it to be fast, it'll be fine

Each TCP connection requires a unique combination of (server port, client port). Your server port is fixed: 80 or 443. They need to use a new ephemeral port for each connection.

You will have 7000 sockets (file descriptors), but that's much more manageable than 7000 ports.

7000 sockets, at any rate, but provided you've anticipated the need, this isn't challenging to support (and nginx is very good at handling large numbers of open sockets)