Do binary blobs pose a potential security threat?
In short: yes.
Binary blobs are by definition not auditable (barring extended reverse-engineering). You don't know exactly what they do, and whether they have backdoors.
One particular binary blob I'd like to highlight is the one in the Intel Management Engine (and the AMD equivalent, the Platform Security Processor). It is a blob that runs on a processor connected directly to your CPU and main memory; it has full access to your OS, hardware and stays on when your computer is turned off but powered.
It is a remote backdoor as its intended use is allowing BIOS management, microcode updates etc. over the internet, by directly accessing your network hardware and communicating with Intel servers over channels encrypted with keys only Intel (presumably) has. It can not be disabled in most newer CPU's.
You can say: "Well, if I can't trust my hardware developer I have no physical security and I'm screwed anyway", but why go for hardware you know has backdoors that can be used by Intel and the American government, and which will potentially one day be exploited by others?
Are all recent computers compromised by this?
Yes. Every i3, i5 and i7 processor is. Every modern Intel processor. Every modern AMD processor. A back-of-an-enveloppe calculation tells you that's at least 300m active computers (according to Wikipedia there are about 433 million computers on the internet in the developed world).
I don't know about you, but I find this deeply disturbing.
The first person to find an exploit in the Intel Management Engine will make in the order of 10's or 100's of millions selling the exploit, or will be able to compromise 10's or 100's of millions of computers in a day.
So is the Librem a good choice for privacy and security?
No. It comes with Intel ME and some other not-nice stuff (BIOS is a blob, microcode upgrades come as signed blobs (so you can't use your own), the video BIOS is a blob). It does have that nice hardware kill-switch, but that can be substituted for with an external network dongle and a piece of tape on the camera.
The OS looks OK, though I don't see what makes it better than, say, Kubuntu or some other user-friendly Ubuntu/Debian/Arch distro, and this is pretty easy to install on any laptop (though hardware support can still vary a little bit; "best laptops for Linux" on Google seems to have very accessible guides on this).
A guy on Reddit has a full breakdown on the Librem laptop, I thought it gave a very clear overview.
Do any laptops come without binary blobs?
Yes, all FSF certificated laptops. The Libreboot T400 is the one which has the best specs of all 3. Yes, there are very few FSF certified laptops, full list here.
Another place to look could be the list of supported Libreboot hardware, but then you'd have to flash your own bootloader and deal with the other firmware on your laptop yourself.
I personally would recommend you consider the Libreboot X200 or the Libreboot T400 if you're going for a privacy-conscious laptop. They're made by The Ministry of Freedom Ltd., which features Leah Woods, Libreboot's main contributor.
These two run laptops Libreboot which does not use any binary blobs (Libreboot is a deblobbed distribution of Coreboot). In fact, these laptops feature no binary blobs whatsoever, and come with instructions on building and flashing your own firmware.
In addition to coming with the full sources of all firmware and having hardware kill-switches for networking and the camera/mic, it features a fully-disabled Intel ME (Management Engine).
You will be getting a lot less performance (Intel Core2Duo upgradeable to Core2Quad) for your money, but it is the one of the only options available that ticks all privacy and free-as-in-freedom boxes.