Sorry, I would have answered the question, except well, the feed doesn't work very well, and I've been doing a lot of work lately. It slipped through my net.
I suppose the real question is 'Just how much are you willing to spend?' and 'So what are you planning to do in the long run with it?'
If you're not willing to drop more than say 500ish for the entire setup (Namely, you're not willing to either pick up a SSD, a SAS or two and controller, or make a RAID0 SATA2/3 mechanical drive set), or if Youtube/basic broadcast is it, then you're better off sticking to something that compresses prior to writing to the disc, namely something that compresses H.264 the moment you hit the record button.
The Hauppage, the Avermedia and others can cover this requirement.
Just note, that if you don't like the idea of 'precompressed' stuff, or you're doing higher quality work, a hardware compressed may disappoint you, and you stuck with some of the quirks (Some of them will reencode the 720p59.94 at 720p60, causing some oddities as an example). What you see is ultimately what you get, and you get what you paid for. Sadly, 300 dollars isn't all that much in the video processing world.
The BM Intensity Pro does not compress via hardware (It's up to you how you want your video to record) but the requirements aren't actually 'You need a Thunderbolt/USB3 connection' (as a PCI-e card exists) and more 'Do I want to capture completely uncompressed video, and fix it up how I like?'
Namely, if not you're prepared to stick in a device that can write at least 160MB/s (Namely a SSD/SAS or a couple of SATA3s in RAID0 format to get the requirements) you won't get very far. Uncompressed 720p59.94 video requires the ability to write quickly and constantly.
I take this line because of the fact I have a separate PC that can handle all the heavy lifting - If I encode to youtube, I just throw it in a specific setting, then send it. If I need to deliver higher quality video for a friend wanting to work on a quick MAD, I encode it in that, then upload. If I need to deliver pixel perfect raw or lossless for heavy post production work, I record then prepare to suit. If I'm broadcasting live, I set up the PC to encode on the fly and go.
As for my setup, it's a BM Decklink Extreme 3D+ (or 2K I think it's called now, but it can record 1080p60, but im@s PS3 doesn't support that), coupled with a 4x 146.8GB SAS array in RAID0, with a support HDMI split that lets me evade HDCP (I'm not in the US, and consequently not subject to the DCMA).
I also keep around a first generation Blackmagic Intensity Pro (PCI-e) which is kicking close to 4 years old which acts as a second capture screen. It's actually the same card that you can still get now, but I paid about 800 dollars for it as it was considered a entry level production card at the time.
The said setup is expensive, but I've been doing this for a while. If you really have a couple of thousand (or more) you'd probably want to carefully consider and study your options, or talk to people who've done a lot of work.
If you just want a cheap im@s thrill, then stick with the cheaper stuff, but if you find that you want to expand into more detailed work, don't hesitate in sidelining the current recording hardware you first pick up here.