A leased line should also be more secure.
How so? As far as I am concerned, any data which leaves or enters an end user's site which isn't encrypted is totally insecure anyway.
A public address can receive all kinds of attacks and rubbish. A leased line should only receive data from the other end.
As candlerb states, you're confusing a point-to-point private circuit with a leased line. The former is used to connect the LANs at different sites together directly without going over the internet. The latter connects a site to an ISP. It can also be used to connect the LANs at different sites together using a variety of VPN solutions but all the traffic passes over the internet.
I think you're also a bit unclear what a public address is. Unless you're connected with what's called Carrier Grade NAT (Hyperoptic uses this), your internet connection will have an exposed public IP address too. If the connection supports IPv6 it will have more than one public IP address.
The main difference between your connection and mine is that I have had a connection with a block of static IP addresses ever since 1998.
When you reboot your router, it may very well get allocated a different IP address. My router doesn't.
The two leased lines I had at home in the past did indeed receive all sorts of rubbish and I have been receiving incoming rubbish constantly ever since 1998. That's why I have always used an enterprise grade firewall - which shows over 60,000 connection attempts blocked last week. I'd have to upgrade to something a bit meatier to take advantage of a 10Gbps connection though. My current firewall only has gigabit ethernet.