I looked over that elgato device and user manual. That site terrible and the user manual is devoid of technical info that would answer a lot of questions.
Basically that device is not unlike the USB/firewire/thunderbolt capture devices that have been around for a decade. They have just put some fancy graphics on the capture interface, allow you clip some media, and included upload functions to social sites.
Assuming you have the newest device, it should be capturing at 1920×1080-60P at 40Mbs/s. The device may actually be able to capture and scale the video to whatever size you input. I would setup you gaming console or PC to output 1920×1080 and capture at the highest bitrate or setting. Then find where on your drive the “Elgato” is storing the raw capture files, and bring those into Vegas. Edit your content in Vegas and export to XDCAM-EX.mfx or SonyAVC.mp4 @20mbs for upload to YouTube. You can upload XDCAM-EX.MFX directly to YouTube, TWICH I am not sure they support this. Do your own tests.
The problem with most of these gaming devices is that they are produces to maximize profits, and save on video codec royalties to other manufacturers. In doing this the engineers develop their own encoder chips that might not produce compliant mp4 files. They cater their software around these differences so you think its the next app down the line that is not working.
If the Elegato source recorded files can be brought into Vegas directly, I would convert all the source material into XDCAM-EX.mxf, or HDCAM-SR-Lite.mxf, or XAVC-HD.mfx for 60P material. Then edit your content from the resulting files, and render as stated above.
“Media Info” is a tool to determine what the Elgato is outputting.
https://mediaarea.net/en/MediaInfo
“Mirillis Action” is another way of doing what you are trying to do and uses .MP4 hardware directly on the GPU in your PC. You do need a supported GPU that has the hardware. NVidia makes “shadowplay” that comes with the GPU, and does the same thing. The editing of the content and uploading is the same as above.