Why do we learn Ancient Greek and Latin using the modern alphabet and not the ancient ones used at the time?

It’s an interesting question, with a boring answer. Because there’s no point.

Let’s break that down though.

1. Right up until the 19th century, the main language being written in Greek script was Ancient Greek; and right up until the 17th, the main language being written in Roman script was Latin. The script hands and typefaces (i.e. “fonts”) evolved over centuries, as memes do, but there was no differentiation felt necessary between Ancient Language font and Modern Language font.

2. People didn’t read Ancient Greek or Latin for primarily antiquarian interest: those were the contemporary languages of the intelligentsia. So they got contemporary fonts; and they couldn’t justify for themselves the additional effort of using an antiquarian script.

3.  The authentic scripts of Ancient Greece had not been standardised, and varied  greatly from town to town. If you’re going to go with a standardised version of Greek script—you have it, and it’s what you print Modern Greek with. (That argument doesn’t apply to Roman with Trajan’s capitals of course.)

4. The notion of antiquarianism and authenticity is really a 19th, if not 20th century thing anyway. Like historically accurate musical instruments. Before then, it would not have even occurred to anyone that a 2500 year old text should be reproduced in a 2500 year old font.

5. People are familiar with contemporary fonts, and not with really old fonts—the odd period  embellishment notwithstanding (such as lunate sigmas). It is *difficult* to read even 500 year old manuscripts; that’s why palaeography is a specialist skill. The shorthand abbreviations commonplace in Greek from the 1000s right up until the 1800s are impenetrable to anyone but specialists. Ditto Latin palaeography.

6. The writing conventions of antiquity are even more challenging for contemporary readers. No lowercase, no spaces, minimal punctuation that doesn’t work the same way as modern punctuation does. Is it worth it?

Papyrologists and epigraphers differ from other classicists who work off mediaeval manuscript copies. The texts that other classicists reconstruct from the mediaeval manuscripts follow modern conventions and punctuation; there’s nothing much in the mediaeval manuscripts’ convention they feel beholden to.

Papyrologists and epigraphers OTOH are working with texts written at the time, not copies; so they are interested in preserving more of the look and feel. Still, usually they put spaces and punctuation in, if not capitals; and only the most fastidious will print two versions of the text—one as it appears on the papyrus or stone, and one normalised. Making sense of the original, unspaced, run-in,  unpunctuated text is the specialist’s job—not the reader’s.

7. There’s a related question you haven’t posed, but which is worth posing: why does noone (outside the Gothic Wikipedia) print Gothic in the Gothic alphabet, instead of the Roman? Why does noone print Old Church Slavonic in Glagolitic, instead of Cyrillic? Because the people primarily interested in the scripts feel it’s part of their patrimony, and don’t feel the need or the inclination to learn a more authentic, but completely different writing system, when it’s painful enough to read the originals to begin with.

7a. Although I’ll concede that  Old Church Slavonic is  printed in old fashioned Cyrillic, looking like what it did before Peter the Great made it look all serifed and Western. That’s an exception to the global trend, witnessed with Greek and Latin.

Is there a reason ‘arse’ is spelled in apparent disregard for its own pronunciation?

If you are indeed asking about arse vs ass, I refer you to What is the difference between the words ass or arse? And butt or booty? 

How was Athens chosen as capital for the Greece?

Ancestor worship.

The first capital of Greece was Nafplio (Nauplia), which was an important port in Ottoman times, while Athens was an insignificant village that attracted the odd Western tourist.

In 1834, King Otto (himself a Western tourist) decreed that the capital of Greece shall be the most important city of Ancient Greece. For after all, Otto was King of the Hellenes.