Dear Abi,
Thanks for your updates and kindness.
May I know whether you just type the Title column value as the keyword in the search box in the list? If yes, the Title column should be indexed by default and searchable. I did a test in our environment with the Title column and I can't reproduce the issue.
As you are using the classic SharePoint UI, may I know whether the search works in the modern SharePoint UI and whether the issue only happens in the specific list? If you do a test in other lists in the site, what's the result? Does the issue only happen in the specific site? If yes, it may be a tenant issue or that a custom result source is configured.
Manage result sources - SharePoint in Microsoft 365 | Microsoft Learn
Reindex List is just to manually request crawling and full reindexing of a list. If some updates or changes are in the list, it will reindex the list and make the updated items searchable. It will not delete any item in the list. So it is safe to perform it.
May I know whether you configure some search settings such as the result source before the issue happened?
If it is convenient, I suggest you set the Title column as the indexed column and then reindex the list again and wait some time to see whether the issue goes away at your side.
The last way can be tried may be to reindex the whole site if reindexing the list doesn't work in your environment. It may take some more time to reindex the whole site.
Manually request crawling and reindexing of a site, a library or a list - SharePoint in Microsoft 365 | Microsoft Learn
If you haven't configured the custom search setting like the search result source and the issue only happens in a specific site, it should be a site crawl issue in your tenant which may need to be fixed by the related team from their side directly. Sorry for that our category may have no enough resources on fixing the issue at our side in that situation.
Given that situation, if you are the administrator, I suggest you follow the steps in Get support - Microsoft 365 admin | Microsoft Learn to open a support ticket to our related team as soon as possible.
About admin roles in the Microsoft 365 admin center - Microsoft 365 admin | Microsoft Learn
How do I find my Microsoft 365 admin? - Microsoft Support
The administrator should be able to open the support ticket from Help & support in the Microsoft 365 admin center.
Then you can select the Phone support or the Email support as follows.

Your effort and understanding is highly appreciated!
Sincerely
Cliff | Microsoft Community Moderator