Yes, accent modification certainly exists.
In the classical bel canto tradition, a specifically prescribed form of pronunciation exclusively for singing is generally required, taught in a very precise way through (lyric) diction classes. Many a tome has been devoted to this subject e.g. Adams (2007) for the major operatic languages of 18th and 19th century opera, Italian, German and French; and LaBouff (2007) for English. Some modifications can be drastic, especially with how consonants are handled; most pertinent for speakers of rhotic accents of English is how the occurrence of [ɹ] is heavily restricted.
The reasons for much of this pronunciation modification in this tradition of singing are to do with the mechanism of singing. The use of airflow and placement in singing bel canto and in choral singing is extremely different to how one speaks in conversation. Clarity in pronunciation has to be balanced against requirements of pitch accuracy and the desired vocal tone; these are what make singing different from speaking, despite using many of the same parts of the body. These musical reasons are explained in some of the answers on the Music StackExchange site.
Other traditions of musical vocal expression have their own requirements for text and pronunciation. E.g. in Peking opera (京剧 jīngjù), there are many special pronunciations required, and it also reintroduces an older phonological distinction to the language (the "round-sharp distinction", essentially the alveolar affricates vs alveolo-palatal affricates, which have merged before /i/ in modern Mandarin).
Most vocal music in the popular tradition is now reliant on the microphone for amplication, with a target vocal tone very different to the one used in choral music or in grand opera, one that in general leads to lower distortion of the vowels. Hence the vocal training for modern pop music places less emphasis on diction.
However, this is not to say that modifications to pronunciation do not exist for popular musical traditions; rather, a greater variety (and some may argue, a greater incongruity) exists. However, such 'accent modification' is actually quite common, and lies on a continuum, as Ben Crystal states in You Say Potato: A Book About Accents:
Think of teenage boys wanting to sound like deep voiced, smooth-talking men, or pop stars from Mick Jagger to Adele ditching their accents while singing in favour of the all-conquering - and globally encountered - American variant.
I would agree that it is the ubiquity of the General American accent in media that has made it the "prestige singing accent" for pop music, rather than the idea that features of General American correspond to 'singing in a way that naturally comes easiest, which happens to be a more neutral way of speaking'. Shifting /ɑː/ to /æ/ doesn't seem to me to be 'easier to sing' from the point of view of vocal production, and certainly going from non-rhotic to rhotic would present some challenges.
Interestingly, sociolinguistic prestige seems to be the driving factor in the adoption of select features of American English in the singing of not only British pop, but also hip-hop from New Zealand. This is even more obvious when loaning from English into songs that are in other languages, e.g. Croatian, Czech or French. Thus, I tend to agree with the following:
The accent people use in their singing is more about the style of music than about where they come from.
However, individual singers can often choose how much modification to add for each vowel (indeed, for each note). It forms part of the musical decision-making process.