Hey, sorry for the tardy reply, unfortunately I'm not too down with the CPU or GPU to know if they'd hit any issues, my old rig used an i7 3770 + 560ti which struggled with 1080/60 output, but chewed through rendering stuff with ease. Looking at the specs below for the two processors they look fairly similar, but some sites show a bit of an advantage with the 3770 -
What is the difference between Intel Core i7-3770 and Intel Core i5-3470? Find out which is better and their overall performance in the CPU ranking.
versus.com
I'm not enough of a PC dude to be able to give more real-life experience of these however.
In my experience with Elgato you're paying a bit of a premium for the brand, about 50% more than actually required in some cases, up to eye-watering crazy increases for their other products.
I started with an Elgato, wish I hadn't bothered and gone straight for the AVerMedia range, if you want to look at 1080/60 as the top end the Live Gamer Mini GC311 is the perfect external go-to capture card. If you want to give room to jump up to 4k you would be best going for the Live Gamer Portable 2 Plus - GC513 - this would be my recommendation for the slight increase in costs to future-proofing compared to the GC311.
Generally speaking streaming at 720/60 looks great, or 1080/30 too, assuming the bitrate output is high enough, but you might need to tweak some things in order to make other components of the stream look good.
There's a balance between your output bitrate, your resolution and frame rate, and the capacity of the PC to handle it all, purely going on the comparisons with my old i7 CPU I would imagine you'd struggle a touch getting decent looking 1080/60 gameplay, but 720/60 should be handled with ease, or 1080/30 should be handled with ease too. Playing at 30fps is far more jarring and tiring on the eyes than watching a stream at 30fps so don't worry too much about that, I see too many people pushing 1080/60 with a low bitrate output so the content looks gash, they'd be much better placed to drop to 1080/30 or to double their bitrate.