Found the problem. The previous theme was using page templates (specifically, no_sidebars.php) and since the new theme (and twentyfifteen) do not, it was looking for that no_sidebars.php template on save and just giving a silent error when it wasn’t found. And so the meta fields were not being saved.
All I had to do was set all the Pages back to “default” in the database. The following did the trick:
UPDATE wp_postmeta SET meta_value = replace(meta_value,'no_sidebars.php','default') WHERE meta_key = '_wp_page_template';
