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.