The runes have been around since the 2nd century AD and would've very widespread when the Goths converted to Christianity around 376 or 390 AD.

The current archaeological record does not support the widespread use of runes among the Goths. Most finds have been found in Southern Scandinavia, England and Southern Germany. The amount of Runic finds linked to the Goths or those found in historically Gothic territory are very few.

Apart from that, Roman (Tacitus, though he speaks of "signs" not explicitly of runes) and medieval (the Poetic Edda) sources seem to indicate that the primary use of runes was not to produce literary works or administrative texts, but that they were used in rituals and as protective spells. That is, among the Germanic peoples in the West, no such sources explicitly concern the Goths.

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When they converted rather than writing the bible in their own writing system, the runes, they instead opt for a new one and start writing the bible from there.

The Runic alphabet derives from an earlier Mediterranean alphabet. Present scholarship is still divided about whether this was an Old Italic, a Latin, an Etruscan or a Greek script (or various combinations) but overwhelmingly in favor of a Mediterranean origin. In other words, Germanic runes were not a singular Germanic invention ex nihilo and not any more 'native' than the Greek-derived alphabet of the Goths or, for that matter, the Phoenician-derived alphabet of the Greeks themselves.

(...) do they see it as "unchristian" therefore they distance from their pagan past.

We have to remember that the Vulgate was written in the same alphabet which adorned the pagan Temple of Jupiter in Rome and that the alphabet of the New Testament is the same of that of the Iliad. The ancients took no real issue with this.

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Wouldn't it be easier to write the bible in a script the converts are already familiar with? What was the reasoning behind the switch?

Today, runes are often though of as just another alphabet. I mean, you can easily go online and transliterate your name into runes, so indeed the question may rise why the Goths didn't just do the reverse when they first encountered Christian missionaries and their strange book?

The runic alphabet isn't particularly hard to write, but, paradoxically, it's not especially easy to read. This is because it has certain graphic and orthographic conventions which are typical for early stages of literacy and not particularly suitable for writing long, complicated texts. For example, there are no rules concerning which direction you are supposed to write. Left to right, right to left, downwards, upwards ... it seems to have been a matter of personal preference. Similarly, it was common to invert certain letters at will and to make words and sentences run together uninterruptedly. Lastly, if a word ended with the same letter that the next word began with, it was typical to not write it twice.

In other words, a spoken sentence like "he ate two oranges" could be written as "segnarowtetaeh".

Now this is fine for small inscriptions, of which there are many ... but less workable for longer texts, of which basically none are extant prior to the 4th century. Even the longest known runic inscription, the Rök stone in Sweden from the 9th century only has about 750 characters.

Hence when Ulfilas (who was already literate in Latin and Greek) compiled the Gothic bible, the advantage of a script based on the Greek alphabet, which had long since been used for long literary works, would have been obvious.

There were additional advantages in this specific situation, as Ulfilas was making a bible translation, specifically the New Testament; which was originally written in (Koine) Greek. By using a Greek-based alphabet, it would have been easier to transliterate the specific (seemingly untranslatable) Christian terminology (which was in Greek) into Gothic. This was done extensively, because the Gothic bible contains a very significant amount of Greek loanwords.