There are antennas out there with less frequency dependence.  They have spirals or fractal shapes or an array or more.  This is where the math and physics get so heavy that they tend to use simulation tools.  But I don't know if those kinds are used for this SDR application.

An SDR will not sample 0 to 5 GHz all the time. There is still a down converter ("tuner") preceding the sampling ADC which then samples the mixing products at a couple (tens) of MHz (the sample frequency directly determines the complex baseband bandwidth).

You will still need a tuned antenna to have good reception, but you can build fairly high bandwidth antennas (discone for example). AFAIK you only need to be "highly tuned" when you're transmitting and want the antenna to radiate as much energy as possible and minimize reflections back into the transmission line.

I'm glad I'm not the only one who's been stumped by this ;)

I think radio isn't quite as tuned-antenna-specific as we'd been led to believe... just think about old boomboxes, those things had one antenna to pick up all the channels... AM and FM... and as I recall, they seemed to work best when the antenna was fully-extended ;)

Most AM/FM radios have separate antennas for the two bands. The extendable antennas are usually roughly the right length to be a match for roughly the middle of the FM broadcast band (88 MHz to 108 MHz). The AM antenna usually consists of a ferrite rod with a few hundred turns of magnet wire around it. It is a compromise, as an antenna actually tuned for the AM broadcast band (0.54 MHz to 1.7 MHz) would need to be several hundred feet long to be properly tuned. The same is somewhat true for the FM antenna, as using a single antenna length across 30 MHz of VHF spectrum is a compromise.

While anything conductive can be used as an antenna, for both receiving and transmitting the most efficient design is going to be an antenna cut to the appropriate length for the given frequency. This will usually either be 1/4 or 1/2 of a wavelength, with adjustments made for the velocity factor of the material the antenna is made of, and the impedance of the feedline. 

Any wide-band antenna will be a compromise, and will not be as efficient as a frequency specific one. There are advantages and disadvantages of either type, and most radio hobbyists end up needing to make a decision on what type of antenna to use based on individual need/desire/limitations.

As for SDRs having huge bandwidth, what I said about antennas is probably also true about SDRs. They can receive over a huge range, but not terribly efficiently. My own tests back that up somewhat, though I was pleasantly surprised at how sensitive those $20 SDRs can actually be (good, but not on par with my ham gear). Another issue with a wide receive is that adjacent strong signals can and will drown out weaker signals, which are usually the ones that a hobbyist is trying to receive. The most efficient and effective receiver and/or transmitter is going to be one with narrow filtering and careful tuning for the frequency range it is designed for. Multiband transceivers usually employ multiple selectable filter paths within the radio, allowing them to be as efficient as possible on all bands covered by the radio. 

A lot of SDR hobbyists (myself included) have added filters to their SDRs to make them work better on the desired frequency.

Now this dude... he's a wealth of knowledge on radio-stuff and has some pretty good humor too :)

Thank you, kind sir, though I think you are giving me more praise than I actually deserve. 

Digital radios can be very sensitive which means that they can pick up signals that have been degraded significantly by poorly tuned antennas.

* Full disclosure, not a radio guy but... The antenna is still just as important for SDR. That digital filtering still happens but on the frequencies targeted from the tuned antenna. Where did you read it only needs one length of antenna?