I started my career as an archaeologist before I ended up in mathematics. I say this in order to emphasize that I am interested in trying to understand how people thought in the past—the usual "text" that an archaeologist reads is the collection of artifacts which are left behind, but there is also a very active field called Historical Archaeology which seeks to associate historical records with a "ground truth". From the point of view an "archaeologist" (or historian) of mathematics, I think that texts such as Loney may be interesting, and well worth reading. However, I would recommend against using such a text in an introductory class.
The cons which you mention are significant, but there are another couple of issues which should give you further pause:
Looking over the table of contents, it appears that much of the focus is on computation. For example, starting around page 106, there are many pages spent on computing the sines and cosines of angles using the angle sum and half-angle formulae. My experience is that modern exposition concerns itself more with the formulae themselves (as these recur in calculus), and largely elides explicit computation. Such computation is, perhaps, useful as an exercise, but a CAS can typically do the job faster and more accurately. In general, my preference would be to use a book which places much less emphasis on computation or which emphasizes the way in which modern computers can aid computation.
There are a lot of topics in that text which are kind of archaic, or which are wholly inappropriate for a modern precalculus class. For example, most of Chapters X and XI are not relevant in a modern classroom (there is no reason for students to be taught how to read a log table, for example). Much of Chapter XV seems to focus on aspects of geometry which, for better or worse, are not typically part of the standard US precalculus curriculum (some of it might show up in the high school curriculum, but a lot of it isn't part any standard curriculum prior to upper division courses in geometry (or, perhaps, math competition prep courses)). These topics certainly have some interest, but if the goal is to prepare students for a standard calculus curriculum, then they don't do any favors to the students.
And then there is Part II. Almost nothing in Part II is part of the precalculus curriculum (and nothing in Part II is mentioned in the course description in the question). It starts with series representations of the logarithm and exponential (though using notation which is difficult to parse by modern standards—the funky factorial and the lack of Sigma notation, for example), moves on to a couple of limits, then jumps into complex analysis. The entire second half of the book is, in most modern classrooms, covered in courses on calculus and complex analysis. It doesn't belong in a precalculus class.
The fact that Part II is inappropriate for a precalculus class isn't a big deal—Part I certainly contains enough material for a semester—but I think that it might be better to select a book which is more narrowly focused on the topics which you actually need to cover.
In the question, it is noted that the language "may sound archaic to some students". I think that this fails to capture the magnitude of the problem—I think that students are likely to bounce hard off of a topic which is presented with unfamiliar notation (and, by the way, unfamiliar notation which they won't ever see again) written in a form of English which is distinctly old fashioned. They inconsistent typesetting also does the book no favors (but now I'm being catty).
I am struggling for an analogy which is not hyperbolic, but I think that the following works: you don't teach students Russian by asking them to read Война и Мир (War and Peace) or Евгений Онегин (a novel by Pushkin) in the original pre-reform language right out of the gate. Get your students to read something from the 20th or 21st Century, first (maybe something like Один День Ивана Денисовича—the language is modern and pretty accessible, and it is still a classic). Anyone who wants to become a scholar of Russian literature should probably get familiar with pre-reform works eventually, but that isn't the place to start. Start with modern Russian, and work back.
Similarly, students of mathematics should start with a modern presentation and then, if they want to study the history of mathematics, start trying to tackle the classics (presuming that Loney is, indeed, a "classic", and not simply "old").
Basically, my recommendation would be to find another text. I don't think that Loney is appropriate for a modern introductory audience. At best, you might use it supplement the course (for example, one of my critiques of the text, above, is that if over-emphasizes computation, at least by modern standards; however, there appear to be a lot of exercises in the text, which might prove useful). Moreover, there is nothing wrong with reading a book such a Loney's and drawing inspiration from it (personally, I have gotten a lot of milage in my precalc classes out of Gelfand's Method of Coordinates and Klein's Elementary Mathematics from an Higher Standpoint).
Unfortunately, I also don't have a lot of advice about which book you should use. There are a lot of books out there with titles like Precalculus (with Trigonometry!) and Trigonometry for the Precalculus Student and whatnot. Most of these books are 1,000 pages long, weigh 10 lbs., and cost $200+. They make good doorstops, but are otherwise too dilute and broad to be of much use. I'm also not a huge fan of the Schaum's Outlines as course texts. They are useful for exercises, but leave a lot to be desired vis-à-vis exposition (of course, that is kind of the point—they are outlines).
Perhaps consider one of the open source texts that are out there? For example, the OpenStax Algebra and Trigonometry or the Open Textbook Library's Trigonometry. I suspect that these books will suffer from many of the same problems as the $200+ tomes, but at least they are free. :\
