I think a huge reason as to why many people believe that being gay is a 'choice' is because of the lack of education on homosexuality.
After all, when I really think about it, it doesn't help when schools don't talk about ALL the different types of sexualities – I know my school didn't (apparently there is only one sexuality… lol). Heck, I didn't even know there was a sexuality called 'asexual', until I researched my own sexuality through Wikipedia, and it came up with many sexualities.
It also doesn't help when David Attenborough won't educate society on the fact that there are gay animals, and that homosexuality exists in nature.
It also doesn't help when the media refuses to talk about gay-related issues and/or news. The other day for example, this program over here in New Zealand, "7 days", talked about the protest in Russia of a man nailing his genetals to a brick (being fully naked too) to make a point about the anti-gay laws, but the media never said WHY he did it, they only said it was a 'protest'. How interesting. I experienced the same discrimination another day when a tv program wouldn't give the reason why whatever happened, happened (because it was gay-related), they only said that it just, well… happened.
Also when you look at places in the world that don't have access to (or are silenced/blocked from) any education on homosexuality, like Uganda, or, what the Russian government is trying to do, by literally gagging anyone telling the truth in that being gay is natural and simply another sexuality, you begin to realize that it's because those places don't have the education or tools to gain a new understanding, and therefore are stuck in the X century.
In various books by David Kinnaman, such as 'You Lost Me' and 'Unchristian', there is good evidence and research to suggest that TECHNOLOGY has helped advance our understanding of all sexualities tremendously. And when you think about, who were we to ask all these questions we ever had, before Google came along? Now, whenever we have a question, in this generation at least, all we have to do is go onto the internet and type away, or swipe our finger across a screen. This generation is also quite sceptical in the sense that, rather than taking what is said for 100% fact, they go and do their own research and come up with their own conclusions. An example of this is how more and more people for example, have left the church or their faith entirely, because of some in their own faith community who have been hypocritical and judgemental towards the gay community, especially after someone they have known and trusted their entire life has come out to them, and knows that they wouldn't wilfully choose to live a life full of adversity or tribulation (and furthermore, that there is no 'gay agenda' as some nut jobs seem to believe).
All that to say (TL;DR), I believe the main reason why people still think being gay is a choice, is because they haven't had 1) ANY education or 2) the RIGHT education – on homosexuality.
Edit: What Jamesn said reminds me of another point I feel the need to make. Many people (including my own parents) just took the bible literally word for word, but never truly examined it to the extent that I ultimately had to, when I was dwelling over the clobber passages literally daily in my head (although I have since had closure to all that after reading 'The Children Are Free: Re-examining the biblical evidence on same-sex relationships' by Rev. Jeff Miner and John Tyler Connoley). I found that, if we took the bible literally, we need to have a SERIOUS talk, because if we are to stone people for being gay (which is a sexual orientation that you cannot change, and therefore one can only ask – how can that be a sin? That's like saying it's a sin to be born with black hair… please…), then we, according to the bible, should also stone children for disobeying their parents, stone women who aren't virgins on their wedding night, not wear anything with two different fabrics, not shave, and the list goes on and on.
I would encourage anti-gay Christians to seriously do some research on the bible, because if they did, they would find out that Sodom and Gomorrah wasn't destroyed because of gay people, in fact, it explains why in Ezekiel 16:49. This is the sin of Sodom:
49 “‘Now this was the sin of your sister Sodom: She and her daughters were arrogant, overfed and unconcerned; they did not help the poor and needy. 50 They were haughty and did detestable things before me. Therefore I did away with them as you have seen."
And in the story of Sodom and Gomorrah, attempting to rape the two angels disguised as men would count as detestable, would it not? This was not about homosexuality at all, it was about sexually violating guests (rape), and literally going after strange flesh (sex with angels). Nowhere is homosexuality mentioned as being the result of Sodom's destruction (and if it is, then most likely it is one of the bibles that have had words added to them – because none of the original bibles had a homophobic translation of the account of Sodom's destruction), and furthermore, let's wrongfully assume that all the men of Sodom were gay as many people suggest they were… why would Lot offer these supposedly gay men, two virgin women? I may not be Albert Einstein, but I AM gay, and I know that when you're a gay man, it usually means that you AREN'T INTERESTED IN WOMEN SEXUALLY. Anyway… I shall leave my own revelation of Sodom and Gomorrah here and allow people to question all this for themselves and come to their own conclusions.