What are the reasons that researchers in computational geometry prefer the BSS/real-RAM model?
First of all, computational geometers don't think of it as the BSS model.  The real RAM model  was defined by Michael Shamos in his 1978 PhD thesis (Computational Geometry), which arguably launched the field.  Franco Preparata revised and extended Shamos' thesis into the first computational geometry textbook, published in 1985.  The real RAM is also equivalent (except for uniformity; see Pascal's answer!) to the algebraic computation tree model defined by Ben-Or in 1983.  Blum, Shub, and Smale's efforts were published in 1989, well after the real-RAM had been established, and were almost completely ignored by the computational geometry community.

Most (classical) results in computational geometry are heavily tied to issues in combinatorial geometry, for which assumptions about coordinates being integral or algebraic are (at best) irrelevant distractions.  Speaking as a native, it seems completely natural to consider arbitrary points, lines, circles, and the like as first class objects when proving things about them, and therefore equally natural when designing and analyzing algorithms to compute with them.

For most (classical) geometric algorithms, this attitude is reasonable even in practice.  Most algorithms for planar geometric problems are built on top of a very small number of geometric primitives: Is point $p$ to the left or right of point $q$?  Above, below, or on the line through points $q$ and $r$?  Inside, outside, or on the circle determined by points $q,r,s$?  Left or right of the intersection of segments $qr$ and $st$?  Each of these primitives is implemented by evaluating the sign of a low-degree polynomial in the input coordinates.  (So these algorithms can be described in the weaker algebraic decision tree model.)  If the input coordinates happen to be integers, these primitives can be evaluated exactly with only constant-factor increase in precision, and so running times on the real RAM and the integer RAM are the same.

For similar reasons, when most people think about sorting algorithms, they don't care what they're sorting, as long as the data comes from a totally ordered universe and any two values can be compared in constant time.

So the community developed a separation of concerns between the design of “real” geometric algorithms and their practical implementation; hence the development of packages like LEDA and  CGAL.  Even for people working on exact computation, there is a distinction between the real algorithm, which uses exact real arithmetic as part of the underlying model, and the implementation, which is forced by the otherwise irrelevant limitations of physical computing devices to use discrete computation.

Within this worldview, for example, the most important open problem in computational geometry is the existence of a polynomial-time algorithm for linear programming.  No, the ellipsoid and interior-point methods don't count.  Unlike the simplex algorithm, those algorithms aren't guaranteed to terminate unless the constraint matrix happens to be rational.  (There are combinatorial types of convex polytopes that can only be represented by irrational constraint matrices, so this is a nontrivial restriction.)  And even when the constraint matrix is rational, the running times of those algorithms aren't bounded by any function of the input size (dimension$\times$#constraints).

There are a few geometric algorithms that really do rely heavily on the algebraic computation tree model, and therefore cannot be implemented exactly and efficiently on physical computers.  One good example is minimum-link paths in simple polygons, which can be computed in linear time on a real RAM, but require a quadratic number of bits in the worst-case to represent exactly.  Another good example is Chazelle's hierarchical cuttings, which are used in the most efficient algorithms known for simplex range searching.  These cuttings use a hierarchy of sets of triangles, where the vertices of triangles at each level are intersection points of lines through edges of triangles at previous levels.  Thus, even if the input coordinates are integers, the vertex coordinates for these triangles are algebraic numbers of unbounded degree; nevertheless, the algorithms for constructing and using cuttings assume that coordinates can be manipulated exactly in constant time.

So, my short, personally biased answer is this: TTE, domain theory, Ko-Friedman, and other models of “realistic” real-number computation all address issues that the computational geometry community, on the whole, just doesn't care about.