Whether you would or would not benefit from using an eGPU depends on what kind of outputs you are going to use. Regardless, you want to make sure that you don’t have ANY displays connected to the computer's own HDMI or USB-C outputs if you are using an eGPU… otherwise performance will slow down dramatically.
If you are using the graphics outputs of the eGPU only, then you should be fine to use the eGPU and get better graphics performance than with the Mac Mini's integrated graphics. Again, you will have significant performance reduction if any of your outputs from the Mini are being used.
SDI Outputs (such as with a Decklink Duo2)
eGPUs are not advisable if you are using SDI outputs because of the limits of Thunderbolt 3 throughput, which is 22Gbps bi-directional. As an example, I'll assume that we want to have an Alpha Key and Fill output, a Stage Display output, and a Lobby Screen output… all at 1080p60.
In ProPresenter, all content has to be rendered in the GPU. Let's assume the lobby screen is going to be playing some videos. Each video is decompressed within the Mac Mini and those decompressed frames then have to be sent over the Thunderbolt 3 bus to the GPU for rendering. To keep numbers simple we’ll say that’s 6Gbps… Now, you have to bring those frames back to the mini so that they can be sent to the Blackmagic Duo2… again, 6Gbps but this time from the other way.
To elaborate and complete the signal flow…
Outgoing from Mini
Decompressed video frames to GPU: 6Gbps
Alpha Key and Fill: 8Gbps (it’s an RGBA format with the alpha so higher than a single stream, but lower than two)
Stage Display 1 to Duo2: 6Gbps
Stage Display 2 to Duo2: 6Gbps
Incoming to Mini
Main video feed key and fill: 8Gbps (again, it’s an RGBA format with the alpha so higher than a single stream, but lower than two)
Stage Display 1 from GPU: 6Gbps
Stage Display 2 from GPU: 6Gbps
You’ve gone over the limit with the outgoing data and thus your video frame rate may suffer. Now, reality is that the data rates are not as high as I've specified here so it may work just fine, but you have very little margin for four 1080p60 feeds.