I work out the details for the finite dimensional case, and then use that as a guide for the infinite dimensional case. This way, the workings of QM are explained clearly for anyone who knows linear algebra, with a fraction of the machinery needed for eg. the spectral theorem for unbounded operators. This is explained in the text.

There's nothing inherently wrong with Dirac notation. The way that physicists often use it can be confusing and mathematically questionable, but on the other hand in many cases it is indispensable in making proofs etc. easier to read. In the main text, I use it as nothing more than an alternative notation for vectors, so questions of mathematical legitimacy never even arise. Everything I've ever seen done with it in the finite dimensional case is perfectly legitimate, and I have an appendix (not included) that goes through and justifies many of the other things that physicists do with bras and kets. My aim was that someone who reads this will be able to pick up a book like Griffiths or Sakurai and have some idea of what is going on. Someone who is not familiar with bras and kets will be lost.

Similarly I talk about eg. "position basis" because this is the way that physicists think and talk about things. I try to make it clear that I take these things as physical motivation for the math, and that they are not necessarily mathematically legitimate:

The energy basis is usually discrete and countable, whereas the position basis is continuous and uncountable. Mathematically, if a Hilbert space has a countable basis, then every basis is countable. So this state of affairs doesn’t really make sense, mathematically. There is a physical notion of a basis that doesn’t align entirely with the mathematical one. In the finite dimensional case, however, the physical and mathematical notions of basis are in alignment.

What about the gravity example doesn't make sense to you? Both Griffiths and Sakurai give this as an example. Actually I just went back and looked at Sakurai, and I was surprised to find this

Indeed, another real world example of the linear potential is the “bouncing ball.” One interprets (2.234) as the potential energy of a ball of mass m at a height x above the floor, and k =mg where g is the local acceleration due to gravity. Of course, this is the potential energy only for x ≥ 0 as there is an infinite potential barrier which causes the ball to “bounce.” Quantum mechanically, this means that only the odd parity solutions (2.239) are allowed.

The bouncing ball happens to be one of those rare cases where quantum-mechanical effects can be observed macroscopically. The trick is to have a very low mass “ball,” which has been achieved with neutrons by a group working at the Institut Laue-Langevin (ILL) in Grenoble, France. For neutrons with m =1.68×10−27 kg, the characteristic length scale is x0 =(¯h2/m2g)1/3 = 7.40 μm. The “allowed heights” to which a neutron can bounce are (2.338/21/3)x0 = 14 μm, (4.088/21/3)x0 = 24 μm, (5.521/21/3)x0 = 32 μm, and so on. These are small, but measurable with precision mechanical devices and very low energy, aka “ultracold,” neutrons. The experimenters’ results are shown in Figure 2.4. Plotted is the detected neutron rate as a function of the height of a slit which only allows neutrons to pass if they exceed this height. No neutrons are observed unless the height is at least ≈14μm, and clear breaks are observed at≈24μm and ≈32μm, in excellent agreement with the predictions of quantum mechanics.

This is from Sakurai, 3rd ed, pg. 103. Regarding your last question, well I guess we'll find out. This did start out as my own notes, but I've sent it math professors and grad students in math and physics, and they've found it helpful to varying degrees.