It's been a long time since email standards have only stored mail on the client.
The latest RFC standard for POP3 (the mail client standard that stores messages client-side) was last updated in 1996. Just about everyone has moved to IMAP, which stores messages server-side, with synching and caching to a client (which also allows multiple synchronous access to a mailbox, so multiple devices can access mail at the same time (e.g. your Mac, iPhone, iPad, etc.)
That's not to say POP3 is dead - some mail providers still support it and, indeed, so does Apple Mail (the client), but for the most part it's considered deprecated.
All is not lost, though. While poorly documented, Apple Mail does have search filtering abilities, you just have to know how to use them
For example,
To see any message containing the words either 'Uncle' or 'Fred': 
if you want to see only messages that contain 'Uncle Fred' use the search syntax:

To see any message from Uncle Fred (as opposed to messages talking about him):

To filter by date, you can add the 'date' tag. To show messages from Uncle Fred from 2022, use:

The date can be narrowed using standard m/d/y notation (or d/m/y if that's your thing):

This will show Uncle Fred's messages from January 2023 only
If you want to use a range of dates, use:

This will show all messages from Uncle Fred from 1/1/2021 through 12/31/2023 - note that the search tag uses '-2024' meaning 'up to 2024', and therefore does not include messages from 2024. To include messages from 2024, just use next year (e.g. 'date:2021-2025'). For some reason, this format does not accept m/d/y dates.
For speed, Apple Mail will use the local cache of messages to display the initial list, reaching out to the server for any additional messages it doesn't have cached, meaning you don't have to save them all on your machine all of the time.
There are also other tags you can use to filter to vs. from addresses, or specifically message content.
So maybe Apple Mail does fit the bill after all. It is also available via your web browser, if that's your preference. I'm not sure why you think it isn't.