This is not an answer that points to a more recent and more accessible account of the triangulability of surfaces, but rather a way to make the account in the first chapter of Ahlfors' & Sario's book more accessible, if sufficient time is available. It should be noted that the proof given by Ahlfors & Sario works for all (connected, 2nd countable) surfaces: compact or noncompact; with or without boundary. I will describe what I found to be the difficulties with Ahlfors' & Sario's presentation and how one can mitigate these difficulties, especially if the learning of this material is by self-study, as was the case for me, and not in the context of a university course. Disclaimer: I am a mathematician but not a topologist.

I found there were three main difficulties, all stemming from Ahlfors' & Sario's terse style of writing. The first difficulty is the absence of any references for background material. I found that the classic, self-contained book, Elements of the Topology of Plane Sets of Points (2nd ed.), by M.H.A. Newman, and the second section of the third chapter of the book, Algebraic Topology, by E. Spanier (for the basics of the theory of abstract simplicial complexes), provide sufficient background. The second difficulty is that almost every sentence resembles the statement of a lemma whose proof is left to the reader. The third difficulty is an intentionally omitted proof of a rather difficult result, "46C".

Regarding the second and third difficulties: after filling in the missing details, I decided to write them up in the form of a list of notes (as opposed to an article). I then created a website, on which I posted these notes. Included in them, is a proof of the result "46C" that relies heavily on the material in the cited book by Newman. Although I did not strive for either optimum mathematical efficiency or elegance, perhaps my notes will be useful to others, in making Ahlfors' & Sario's account more accessible.

While I was at it, I also posted some details for a proof of Schoenflies' Theorem via Complex Analysis; these are details for the presentation in the book, Boundary Behaviour of Conformal Maps, by C. Pommerenke. Note that this proof assumes the Riemann Mapping Theorem, proofs of which are more widely available. (These notes were written before I was aware of Newman's book, which happens to also contain a proof of Schoenflies' Theorem -- a proof that is purely topological in nature. As mentioned in Allen Hatcher's answer, an historical account of proofs of Schoenflies' Theorem, including a new one at the time, appeared in the paper and its errata. A preprint of the paper is freely available here.)

A careful accounting of all the prerequisite material for Ahlfors' & Sario's approach, reveals that it is a very substantial amount. Although such a proof of the triangulability of surfaces can be made accessible to undergraduates, it is not clear whether an undergraduate who embarks on such a project of self-study will still be an undergraduate upon completion of the project.

This is not an answer that points to a more recent and more accessible account of the triangulability of surfaces, but rather a way to make the account in the first chapter of Ahlfors' & Sario's book more accessible, if sufficient time is available. It should be noted that the proof given by Ahlfors & Sario works for all (connected, 2nd countable) surfaces: compact or noncompact; with or without boundary. I will describe what I found to be the difficulties with Ahlfors' & Sario's presentation and how one can mitigate these difficulties, especially if the learning of this material is by self-study, as was the case for me, and not in the context of a university course. Disclaimer: I am a mathematician but not a topologist.

I found there were three main difficulties, all stemming from Ahlfors' & Sario's terse style of writing. The first difficulty is the absence of any references for background material. I found that the classic, self-contained book, Elements of the Topology of Plane Sets of Points (2nd ed.), by M.H.A. Newman, and the first two sections of the third chapter of the book, Algebraic Topology, by E. Spanier (for the basics of the theory of abstract simplicial complexes), provide sufficient background. The second difficulty is that almost every sentence resembles the statement of a lemma whose proof is left to the reader. The third difficulty is an intentionally omitted proof of a rather difficult result, "46C".

Regarding the second and third difficulties: after filling in the missing details, I decided to write them up in the form of a list of notes (as opposed to an article). I then created a website, on which I posted these notes. Included in them, is a proof of the result "46C" that relies heavily on the material in the cited book by Newman. Although I did not strive for either optimum mathematical efficiency or elegance, perhaps my notes will be useful to others, in making Ahlfors' & Sario's account more accessible.

While I was at it, I also posted some details for a proof of Schoenflies' Theorem via Complex Analysis; these are details for the presentation in the book, Boundary Behaviour of Conformal Maps, by C. Pommerenke. Note that this proof assumes the Riemann Mapping Theorem, proofs of which are more widely available. (These notes were written before I was aware of Newman's book, which happens to also contain a proof of Schoenflies' Theorem -- a proof that is purely topological in nature. As mentioned in Allen Hatcher's answer, an historical account of proofs of Schoenflies' Theorem, including a new one at the time, appeared in the paper and its errata. A preprint of the paper is freely available here.)

A careful accounting of all the prerequisite material for Ahlfors' & Sario's approach, reveals that it is a very substantial amount. Although such a proof of the triangulability of surfaces can be made accessible to undergraduates, it is not clear whether an undergraduate who embarks on such a project of self-study will still be an undergraduate upon completion of the project.