Where should I host a (simple) personal academic webpage?

This doesn't sound like your website will have special requirements that an all-purpose CMS couldn't handle. My advice is to go for one of those "all included" packages by a mainstream provider with something like Wordpress, and find a nice-looking template. This is the easiest choice, it will guarantee up-time, and you don't have to spend valuable time to administrate or design the site. It is also not very expensive: Perhaps around 5-10 USD/month, plus a one-time expense for the template (5 USD and up). Two alternatives might be attractive as well:

• Nerdy option: If you are familiar with GitHub, you can use GitHub pages to self-host a static website with Hugo, like so. Once it's set up, it's a breeze to update. Here's an example.
• Thrifty option: Use google sites.

Details:

My institution's personal webpages facility

Pro: Demonstrates academic credentials and shows your affiliation. Con: Not much leeway. Will you be able to use it after you leave the institution?

My own web server at home

Pro: All the control you need. Con: Way too much overhead; perhaps you can't implement what you need after all.

Some for-pay general-purpose webpage/website hosting service

Good balance between control and ease of use.

Some gratis general-purpose webpage/website hosting service

Free comes at a price. Perhaps you'll have up-time issues, ads or no own domain, certainly less control.

Using one of the specialty academic-webpage platforms (which apparently exist)

See here.

My current setup is that I have a personal domain which redirects to my institutional website. I recommend this, as it gives you the best of both worlds -- you get the officialness of an institutional website, but also the flexibility of owning your own domain, and it's cheap.

Getting a domain only costs US$1/month and is very easy. (I use Google domains.) If I need to move my website to a different instutition or decide to use something like wordpress, I can simply change the redirect from my domain.

A personal domain also gives you an email address associated with it. You can port this between institutions and it looks more professional than putting @gmail.com on your papers.

Since I first wrote this answer, I have moved institutions and moved my website hosting to Amazon web services. This is a cheap option (~US$0.60/month), but was a pain to set up.

I think GitHub Pages would be a really good solution for you.

Note:

• GitHub Pages source repositories have a recommended limit of 1GB
• Their URL would be username.github.io
• You can enable HTTPS support
• This is something that github pages are generally used for