Properly conducted longitudinal studies have found that, if anything, the trend is in the opposite direction.
Only two longitudinal and prospective studies have examined changes in sexual identity over time, both of which were conducted among young women (Diamond, 2000; 2003; Sophie, 1986). Although no comparable studies exist on the sexual identity development of males, three longitudinal studies of young men have examined changes in sexual attractions (Dickson, Paul, & Herbison, 2003; Stokes, Damon, & McKirnan, 1997; Stokes, McKirnan, & Burzette, 1993). Taken together, the studies have found considerable consistency, as well as change, in sexual self-identification and attractions over time. For example, among 80 female youths (65% college students, and over sampled for youths who did not self-identify as lesbian, bisexual, or straight), Diamond (2000; 2003) found that 70% were consistent in their self-identification as lesbian, bisexual or unlabeled after two years and 50% were consistent after five years. An additional 15% transited to a lesbian or bisexual identity after two years, as did 14% after five years. Few youths transited from a lesbian, bisexual, or unlabeled identity to a straight identity. Among 216 behaviorally bisexual men (ages 18 – 30 years), Stokes and colleagues (1997) found that over the course of one year, 49% reported no changes in sexual orientation, 34% became more homosexually oriented, and 17% more heterosexually oriented. Clearly, the consistency and change documented by these various research studies must now be understood.
Regarding the quoted graph, Kinsey's report was based on self-selected interviewees and no attempt was made to obtain a representative sample. At the time this was ground-breaking research because merely talking about the subject was considered taboo, but you cannot trust any statistics derived from these interviews because of the number of potential confounding factors.
In particular, the graph shows a huge hump in category #6 at ages 13-15 and the smaller one for category #3, but nothing at that age in categories #4 and #5. This lack of correlation between the different categories strongly suggests some kind of sampling artefact.
Two possible explanations that occur to me are:
This data may have come from Kinsey's interviews with paedophiles, who cannot be considered to have valid opinions on their victim's sexuality.
At the time there was widespread ignorance about all aspects of sex, and sex education beyond a "birds and bees" talk from a parent was very rare. Some of Kinsey's interview subjects may have had homosexual attraction or experiences during adolescence and only later found out that what they were doing was considered taboo. They would then repress any homosexual tendency and report that they had since become straight.
I can't prove either of these theories, but they explain the graph shown at least as well as the theory that homosexuals turn straight as they get older.
Lastly, even if sexual orientation does tend to turn straight with age, this is not evidence against a genetic component because ageing in general has a genetic component.