Lightboxes do not lack the ability to remove the overlay.
You can either remove or design the overlay to your liking.
It is the page background property if I remember correctly.
Yes, this would be one option.
But if you want to be able to have an identical notification system throughout the site, you need to add a container to every page.
Furthermore, light boxes come with some nice functionalities which would be perfect to use as notifications.
How would you go about creating something similar?
Container on each page?
What if you change the styling of the container, do you need to do this on every page?
Well, a lightbox will be perfect for that unless as you say, you need to keep interacting wit the surrounding page.
A Container can work, if it fits your needs well enough, while receiving it’s design from a data collection’s rich content field perhaps.
Or a third option currently available would be to create a custom element.