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Edit: Indeed, it is clear from the preceding pages that by "gluing" Grothendieck means the adjunction space, which he also calls "amalgamated sum". In particular, he says:

It was also clear that the contexts of the most rigid structures which existed then, such as the "piece-wise linear" context were equally inadequate one common disadvantage consisting in the fact that they do not make it possible, given a pair $(U,S)$ of a "space" $U$ and a closed subspace $S$, and a glueing map $f: S\to T$, to build the corresponding amalgamated sum.

There is, of course, no problem with forming adjunction spaces in the PL context. Perhaps Grothendieck was just not aware of pseudo-radial projection or something. End of edit

Besides, there are, of course, Kan sets, which are closed under Hom, but they're they arguably don't belong to tame topology in any reasonable sense because they quickly get uncountable (in every dimension, in particular, there are uncountably many vertices) and even of larger cardinality.

In any case, logicians, who tried to set up Grothendieck's aspiration in a rigorous framework of definability (see Wilkie's survey), do now have the "o-minimal tringulability and Hauptvermutung" theorem, saying roughly that tame topology (as they understood it) is the same as PL topology. Still more roughly (perhaps, too roughly) is could be restated as "topology without infinite constructions is the same as PL topology".

Disclaimer: What follows is probably a bit off-topic for this site, but no more than the original questions, numbered one and two. In fact I suspect that this answer attempts to address just what the OP really wanted to ask ("isn't PL topology useless?") by posting those two lightly euphemistic questions. If there was an active meta thread for closing this question, I'd rather put this answer there.

Some topologists, perhaps the majority, tend to think that smooth and topological manifolds are "present in nature" and are the genuine objects of study in geometric topology, while PL topology is a somewhat artificial, unnatural construct, and matters just as long as it is helpful for the "real" topology. I've heard this opinion stated explicitly once, and I see a lot of this kind of attitude in this thread. In fact I think this philosophy/intuition is sufficiently familiar to nearly everyone that I don't need to elaborate on it. Moreover, I suspect that a lot of people are not even aware that it is not the only possible religion for a topologist, or else they would be more considerate to the heretics in stating their strong opinions.

I'd like to discuss one other philosophy/intuition then, according to which both smooth and topological manifolds are obviously artificial, highly deficient models for what could be "present in nature", whereas the PL world is much "closer to the reality". I don't consider myself a practitioner of this or any other religion; what follows should be regarded as said by a fictional character, not by the author.

1) As is well-known, the predisposition to seeing continuous and smooth as more natural than discrete is historical, following centuries of preoccupation with derivatives and (later) limits. Quantum physics and computer science may be changing the tide, but they don't usually compete with Calculus in a mathematician's education, at least not in the initial years.

Here is a simple test. When you fold a sheet of paper, what is the intuitive model in your imagination: is it a smooth surface (when you look with a loupe at the fold), a cusp-like singularity (generic smooth singularity), or an an angle-like singularity (PL singularity)? No matter what is your subconscious preference, I bet you didn't base it on considerations of individual photons detected by the eye. But you could have based it on your previous experience with abstract models of surfaces, which is not independent of the historically biased education. (Just for fun, I wonder if your intuitive model would change if the paper sheet is folded second time so as to make a corner - which is unstable as a singularity of a smooth map $\Bbb R^2\to\Bbb R^2$, but has a stable singularity in the link.)

2) On a molecular scale, the sheet of paper of course doesn't fit the model of a smooth surface, and although it is arguably not "discrete" or "PL" on a subatomic scale, the smooth surface model isn't restored either. Similarly, as is well-known, Maxwell equations and general relativity (which I guess are among the best reasons to study smooth topology) don't work at very small scales. The problem is that this "imperfection" of matter doesn't usually shake one's belief in "perfect" physical space. But it is perfectly consistent with modern physics (for those who don't know) that physical space is kind of discrete at a sub-Planck scale, as in loop quantum gravity (which is somewhat reminiscent of PL topology!). It is also consistent with the present day knowledge, and indeed derivable in variants of the competing string theory, that a finite volume of physical space can only contain a finite amount of information, as with the holographic principle. (In fact I didn't see much discussion of possible alternatives to this principle, many physicists appear to take it for granted.) I'm getting on a slippery slope, but finite information does not sound like it could be compatible with limits that occur in derivatives (which returns us to MacPherson's program on combinatorial differential manifolds) and especially with Casson handles that occur in topological manifolds.

The fictional character is now saying that his religion teaches him to avoid concepts based on inherently infinitary constructions, because they are likely to be unnatural, in the sense of the physical nature which might simply have no room for them (and even the question of whether it does is not obviously meaningful!). Ironically, this is quite in line with Poincare's philosophical writings, where he argued at length that the principle of mathematical induction is not an empirical fact.

3) The fictional character goes on to say that this is not just the crazy metaphysics that displays the warning, but also Grothendieck with his "tame topology" which inspired a whole area in logic (initiated by van den Dries' book Tame topology and o-minimal structures). Here is a short quote from Grothendieck:

It is this [inertia of mind] which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom "wildness" is a fatal necessity, rooted in the nature of things.

My approach toward possible foundations for a tame topology has been an axiomatic one. Rather than declaring [what] the desired “tame spaces” are ... I preferred to work on extracting which exactly, among the geometrical properties of the semianalytic sets in a space $\Bbb R^n$, make it possible to use these as local "models" for a notion of "tame space" (here semianalytic), and what (hopefully!) makes this notion flexible enough to use it effectively as the fundamental notion for a “tame topology” which would express with ease the topological intuition of shapes.

Grothendieck dismisses from the start PL and smooth topology as possible forms of tame topology, because

(i) they're "not stable under the most obvious topological operations, such as contraction-glueing operations", and

(ii) they're not closed under constructions such as mapping spaces, "which oblige one to leave the paradise of finite dimensional spaces".

I'm not familiar with "contraction-glueing operations", nor is Google. Perhaps someone fluent in French could explain what (i) is supposed to mean? My first guess would be that this could refer to mapping cylinder, mapping cone or other forms of homotopy colimit, but PL topology is closed under those (finite homotopy colimits). As to (ii), there now exists some kind of an infinite-dimensional extension of PL topology, which includes mapping spaces and infinite homotopy colimits up to homotopy equivalence (and hopefully up to uniform homotopy equivalence, which would be more appropriate in that setup). Besides, there are, of course, Kan sets, which are closed under Hom, but they're arguably don't belong to tame topology in any reasonable sense because they quickly get uncountable (in every dimension, in particular, there are uncountably many vertices) and even of larger cardinality.

In any case, logicians, who tried to set up Grothendieck's aspiration in a rigorous framework of definability, do now have the "o-minimal tringulability and Hauptvermutung" theorem, saying roughly that tame topology (as they understood it) is the same as PL topology. Still more roughly (perhaps, too roughly) is could be restated as "topology without infinite constructions is the same as PL topology".

Even if smooth topology will some day be reformulated in purely combinatorial terms, it is highly unlikely that it can be characterized by purely logical constraints. From this viewpoint, smooth topology is primarily justified by its role in applied math and natural sciences, but is no less and no more fundamental than symplectic topology or topology of hyperbolic manifolds.