What are the consequences if a site has the same
<title> tag on all the pages, reporting only the site name?
You run the risk of having your pages flagged as duplicate content. The title of a web page is very important for SEO and if all of your pages have the same title you dramatically increase the chances they will be considered duplicate content. You would have to make sure your pages' content is very different prevent this from happening.
Or better (or worse) no title tag at all?
Worse. You definitely want a
<title> for all of your pages. It's probably the biggest on-page ranking factor for SEO. When a user bookmark's your page it is the default text for that bookmark. It's what is shown in the title bar and/or tabs of your pages. Basically not having one is bad for SEO and usability.
Will that be a little/medium/huge SEO problem?
How will the pages appear on search engines?
Hard to say as I have never seen an untitled page in the search results but that's probably because they all rank too poorly to be found. But I suspect Google would put either a snippet found on the page or something along those lines.
Will fixing it in a later stage be problematic since pages have already been indexed?
No. But it may not happen quickly. It could weeks or months for you to see the changes reflected in Google's search results. And all that time your rankings and click-thrus will suffer for it.
How does it compromise the overall usability/accessibility/experience?
No page title means it is more difficult for users to know what page they are on and what the main purpose of that page.
Is that a "feature" that can be omitted, or it can't even be considered a "feature" but a core element?
<title> tag is a core element.