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If they have old URLs you can use the redirects under "Settings" -> "SEO". Oddly placed, but hey, the functionality is there. And for non-techies I think Weebly is wayyyy better than WordPress.

What happens when the organization outgrows the features offered by Weebly, and wants to move their site to another host?

There is an export site functionality which gives you everything: HTML, CSS, images, uploaded files, etc... I'd never actually tried it until now, so thank you :-)

Sounds like a perfect use-case for weebly. Weebly is much easier to setup than Wordpress.. in fact there is no setup at all

For a website like that, I would recommend a simple Wordpress site on a basic shared hosting provider. It's quick to get set up with the most important information, and then easy to keep updated by multiple non-experts.