Can sensitive nuclei like 1H be exploited to obtain 13C spectra?
We are able to record 1D heteronuclear experiments using these techniques, and absolutely do make use of this behaviour as a commonplace method for measuring 1D spectra of low sensitivity nuclei, such as $\ce{^13C, ^15N}$ and $\ce{^29Si}$. However, in order to get frequency information for the insensitive nucleus, we still need to use what is called a direct-detect method, where the nucleus of interest is actually the acquisition nucleus.

The most common method for sensitivity enhancement uses the general class of experiments called INEPT experiments; Insensitive nuclei enhanced by polarization transfer. The DEPT series (DEPT45, DEPT90 and DEPT135) are a sub-class of the INEPT experiment. 

The signal of some nuclei can be improved by taking advantage of low power 1H decoupling during the relaxation period to allow nOe transfer to occur. This, as you mention, is typically how $\ce{^13C}$ spectra are recorded, and is why the $\ce{CH3}$ peaks ar much bigger than quaternary peaks. This method cannot be used for nuclei that have a negative magnetogyric ratio, such as $\ce{^15N}$ and $\ce{^29Si}$, as the nOe will actually generate a diminished signal. For these nuclei, the INEPT method is the preferred experiment for acquisition.

The INEPT magnetisation transfer step is at the core of most common 2D sequences, such as the HSQC. It provides a much greater signal enhancement than the nOe, although is strongly dependent on correct calibration of pulses.


The theoretical maximum sensitivity gain for nOe is $\displaystyle1+\frac{\gamma_\mathrm{S}}{2\gamma_\mathrm{I}}$
The theoretical maximum sensitivity gain for INEPT is  $\displaystyle\left|\frac{\gamma_\mathrm{S}}{\gamma_\mathrm{I}}\right|$


For $\ce{^13C}$, this equates to $\approx 3\times$ for nOe and $\approx 4\times$ for INEPT. For $\ce{^15N}$, it is $\approx -4\times$ for nOe (yep, negative) and $\approx 25\times$ for INEPT.

Glenn Facey at University of Ottawa has a couple of nice examples of INEPT and DEPT that are well worth looking at.

Both of these classes of experiment (nOE and INEPT) typically rely on interaction with nearby $\ce{^1H}$ signals: nOe is a through space interaction, and INEPT is a through bond interaction. Other nuclear pairs can be used, and many examples such as INEPT using $\ce{^19F-^29Si}$ have been published. The detection of insenstive nucleus environments with no directly attached sensitive nucleus (such as $\ce{^13C}$ quaternary centres) still requires long range coupling, and there are many published long-range INEPT methods.

The 2D HSQC method is an indirect detection method — we get information about the insensitive nucleus by acquiring another nucleus. Equivalent 1D variants of these 2D experiments do exist (so, a 1D HSQC), but they will only give you information about those $\ce{^1H}$ signals that have coupling to $\ce{^13C}$ — they don't provide any information about the $\ce{^13C}$ frequencies, unless you do a selective variant. So, a sel-HSQC will only irradiate a small band of the $\ce{^13C}$ frequency window, and therefore only give $\ce{^1H}$ signals that correlate to a $\ce{^13C}$ nucleus in this band. This is sometimes useful to probe a specfic question of structure, but a very tedious method of stepping through frequency bands of the $\ce{^13C}$ region (which is what a 2D experiment does for you). 

For what it is worth, there are a number of cost-effective ways to improve sensitivity without having to fork out the big bucks for an 800. Choice of probe (X-observe, cryoprobe, microprobe) will offer improvements over increase in magnetic field at a fraction of the cost, and Shigemi tubes and micro cell tubes will also provide significant boosts in  sensitivity.