If you’re using a current gen Mac mini (i.e. M1 chip) then I think your sampling rate is locked to 48 kHz. No matter what file you play, it will always resample to that rate if you’re using the analog out. This would be the main reason to get an external DAC. It will automatically switch sample rates in addition to possibly giving you the ability to play other hi-res formats such as DSD.
Even if a motherboard adhered to the Intel HD Audio standard, some sampling rates won’t be supported. For example, the Realtek ALC1220-VB codec on my motherboard doesn’t support 88.2 or 176.4 kHz sampling rates (I need the latter for a protocol called DSD over PCM).
Will hi-res files sound better? It depends. Some of them can actually sound worse - particularly if you get a ’remastered’ version of an album first released in the 90’s or earlier. These newer versions simply crush the dynamic range which makes the music sound louder, not better. But this is the result of poor mastering, not anything to do with sampling rate or bit depth. Really, it's the engineering that makes the recording.
The good news is that you don’t need to spend a huge amount of money to get good sound. Companies such as JDS Labs and Topping produce gear that is reasonably priced and well engineered. For example, the Topping DX3 Pro+ is $199 and you get a DAC capable of nearly all hi-res formats plus a headphone amp. It also has RCA line outs to connect to your receiver if you wanted.
If you’d like to learn more about digital audio in general, I highly recommend the following links:
24/192 Music Downloads... and why they make no sense
Digital Show & Tell
The 8-bit vs 16-bit blind listening test