Also not my field, but...

1) Define "few"? There are many available, already for homotopy methods. See http://www.issac-conference.org/2010/assets/SoftwareDemos/VerscheldeFinal.pdf

"RELATED SOFTWARE: Homotopy continuation methods for polynomial systems are implemented in various computer programs, listed in alphabetic order: Bertini [1, 2], CONSOL [14], HOM4PS-2.0 [9], HomLab [18, Appendix C], NAG4M2 [10], PHoM [8], POLSYS PLP [26], and POLSYS GLP [20]. Both POL-SYS PLP and POLSYS GLP are part of the development of HOMPACK [24, 25]. Polyhedral homotopies need mixed volumes, computed by DEMiCs [13], Mixvol [4], and MixedVol [5]."

As Wikipedia noted, Maple has (at least) two solving methods. Mathematica also has some.

2) They are perhaps not easy to build (problems with numerical errors, tracing homotopy paths), but this is not the critical factor.

3) One of the main "markets" is robotics. You can note that some of the notable authors in the field (Wampler most obviously) have jobs either directly in or related to (such as part-time consultant) the car industry.

See also (Wampler and Sommese) http://dx.doi.org/10.1017/S0962492911000067

"While systems of polynomial equations arise in fields as diverse as mathematical finance, astrophysics, quantum mechanics, and biology, perhaps no other application field is as intensively focused on such equations as rigid-body kinematics. This article reviews recent progress in numerical algebraic geometry, a set of methods for finding the solutions of systems of polynomial equations that relies primarily on homotopy methods, also known as polynomial continuation. Topics from kinematics are used to illustrate the applicability of the methods. High among these topics are questions concerning robot motion."

The slides of Verschelde's 2003 talk also have some applications listed. http://homepages.math.uic.edu/~jan/Talks/cimpa_first.pdf

4) The market is not cornered, in particular methods that exploit sparsity are of high interest. For instance, in the Math Reviews of Li's 2003 survey, it is noted that half the 100 pages are involved with sparsity. http://www.ams.org/mathscinet-getitem?mr=2009773

"More than half of this almost 100 page survey is devoted to the exploitation of sparsity, as most applications give rise to polynomials which have fewer monomials with nonzero coefficients than general polynomials of the same degree. Quite often, sparse systems then also have far fewer roots compared to systems of general polynomials of the same degree. Any homotopy continuation method which is based only on degree bounds will likely have to trace far too many diverging paths to find all solutions of a sparse polynomial system."

See also Section 10 of the above Wampler and Sommese paper, for the types of improvements that are being made (or would like to be seen).

5) Yes, though Wikipedia's information here ("25") is not exactly reliable.
For instance, see section 8 of the aforementioned Wampler and Sommese paper (http://dx.doi.org/10.1017/S0962492911000067):

"To put some of this in perspective, consider a sequence of polynomial
systems arising from discretizing the Lotka–Volterra population model that
is presented in Hauenstein et al. (2011). HOM4PS-2.0 and PHCpack both
implement the polyhedral homotopy method for solving sparse polynomial
systems, while Bertini implements regeneration (discussed in Section 10.2)
for solving such systems (Hauenstein et al. 2011). For these systems, the
number of solutions over C is the number of paths tracked by the polyhedral
homotopy method. Even so, the polyhedral homotopy becomes impractical
as the number of variables increases due to the computational complexity
of computing a start system via the mixed volume. For example, consider
solving the n = 24, 32, and 40 variable instances of the Lotka–Volterra
model using a single core running 64-bit Linux. Here the number of non-singular isolated solutions equals $\sqrt{2^n}$ (which equals the mixed volume). For
the 24 variable system, PHCpack took over 18 days while HOM4PS-2.0 and
Bertini both took around 10 minutes. For the 32 variable system, PHCpack
failed to solve the system in 45 days, HOM4PS-2.0 took over 3 days and
Bertini took roughly 5 hours. For the 40 variable system, PHCpack and
HOM4PS-2.0 both failed to solve the system in 45 days, but Bertini solved
the system in under a week. Since regeneration is parallelizable, we also
solved the 32 and 40 variable systems using 64 cores (8 nodes each having
dual 2.33 GH Xeon 5410 quad-core processors). In parallel, Bertini took
less than 8 minutes and 4 hours to solve the 32 and 40 variable polynomial
systems, respectively."

In this example, Bertini thus found $\sqrt{2^{40}}$ isolated solutions, somewhat larger than the Wikipedia bound of 25. Of course the system should be "special" (likely meaning sparse in some sense) to deal with a large Bezout (or mixed volume) number, but many interesting systems do have particular properties. (The same is true for Gr\"obner basis methods, that the generic behavior might not be very relevant to any specific given instance.) I don't know if by "automatic" you want to be able to solve generic systems of a given Bezout number efficiently, probably that is a different question.