This would be a very ineffective population control method.
How many people are Gay?
11% of Americans have acknowledged same sex attraction. That doesn't make them Homosexual, but it means they are 1 or more on the Kinsey scale.1.7% and 1.8% of Americans are Homosexual and Bisexual respectively. That means that 3.5% of people are Homosexual, and 7.5% are not Gay but have had some same sex attraction.
Come to the Gay side, we have cookies
Despite what right wing media outlets might have you believe there is no large scale gay propaganda (at least telling people to become Gay, instead of just tolerance) in the media. To start, you would have to convince 90% of the 1.8% of the population that are Bisexuals to be Homosexual, since ~90% bisexuals in relationships enter heterosexual relationships. this effort would reward you with 1.6% of the population being turned gay. You would then theoretically convince all the people who are in the 7.5% group (people who have had some same sex attraction but are not Homosexual or Bisexual) to become Gay. This incredible effort result in a total of 9.3% of the population being Gay who were straight before.
Next, the actual impact of becoming Gay, that thing we worked so hard for. Well, according to Swedish numbers, while Gay people have nearly no children, Lesbians only have about half as many children. This number might stabilize, but lets assume that this will stay this way. So this would effectively only cut down on the amount of children that the new Gay population produces by a factor of 4. This means your plan would cut down on the number of people having children effectively by 7.0%.
It's okay to be (not) Gay
Finally you try to convince straight people to be Gay, and you have a problem. Best we can tell it isn't chemicals in the water, left wing TV, or feminism that makes people gay. We already converted all the people who have even acknowledged that they have had gay attraction, so now we are trying to convince people to be gay who have never even though about being gay. Furthermore, there are people out there who for one reason or another, being straight is part of their identity (The ladies man, the girl who is popular with the boys) pushing them is just going to make them more defensive of their straight personality.
So, we put out a massive media circus, broke up on going straight relationships with Bi people, and have got every single person who has ever had a Gay thought into a Homosexual relationship. What do we get?
7.0% reduction in birthrate
This is so low that most people wouldn't count it as a major contributor to the problem. The drop in fertility from 2014 to 2018 in America is greater than this. Video games and drinking less alcohol likely have a larger impact on sex and therefore birthrate. This also assumes that we succeeded in our plan by every possible metric. This is after we spent what could come out to Trillions to get this to occur.
What is actually effective?
Do what Japan does and put more of an emphasis on working, and pump up video games and media in the sparse free time people have. According to this graph a single year of extra education for all women would have the same effect on fertility as your Gay plan, and you get a more educated workforce. Other things you can do is give tax breaks to married couples with no children. Reduce climate action to increase the dread of bringing a child into a doomed world.