Set it up as a Global Source, and hide it behind your title card/splash so it's started up and running the entire time you're streaming. The pause is OBS turning on the hardware, and waiting for it to be ready and sending video. Elgatos seem to take a bit to autodetect and set up when you first switch them on, especially the USB 2.0 ones. Going the Global Source route, you'll probably want to have your console muted or switched off until you change scenes to one that doesn't have the global source in it (so it won't make noise on-stream, and the EG can do its auto-detection routine away from your viewers' eyes; it won't be shown, but since it's Global it'll be always-on and ready to go until you stop streaming).
Hardware upgrade, my go-to cap card is the Micomsoft SC-512N1-L/DVI. Best on the market at present for livestreaming purposes, but $330 can be a bit steep for some.
Can get one from SabrePC as the Yuan SC512 Single-channel for $200, but they seem to have trouble keeping them in stock reliably. Also doesn't come with the passthrough daughtercard (use splitters) or the HDMI->DVI adapter ($7 on Amazon, cheaper on Monoprice).
