That depends. Society has become more polarized in the last ten years especially, which I think is a result of the "tolerant" people actually becoming far less tolerant themselves toward those with opposing views. Various governments and educational establishments have now decided they too are going to enforce this tolerance, but you can't really force tolerance or understanding from anyone, that's a fact. If you start punishing people, forcefully educating them or threatening them, you don't get love or understanding in return.
So if that trend keeps up, no, I don't think we're going to be able to force more understanding or tolerance with the barrel of a gun, because that's human nature. And to be quite fair a number of policies of the establishment that include but aren't limited to LGBT matters are causing wider issues in society that are also affecting public attitudes. Mass immigration for example - there could be a direct clash between, say, individuals in a majority Muslim area of a city and the right of LGBT citizens to host a pride march through the area. While progressive groups may think they're helping the LGBT cause on the one hand, as well as the causes of immigrants, I expect they'll soon see that certain cultures and groups that they also encourage are not compatible with LGBT matters on the whole, and we will suffer because the priority appears to be keeping those immigrant communities happy and calm, not us. I see that as an impending problem of some importance in the future of certain areas of the UK and other Western countries. If we end up with increasing numbers of immigrants from certain cultures and they send individuals to become councillors, judges and politicians in the coming decades, then I don't see understanding increasing at all for the likes of us... because notably these new communities seem to keep their original values more intact than choosing to discard them in favor of ours.
I'm not expecting the world to be a much more welcoming place in the future. At best it will be divided on opinion with places we should and shouldn't go for our own safety, at worst attitudes toward us could regress to where they were in the 40s and 50s, or god forbid, the Middle East and Africa.
A lot of folks seem hopeful about it but it seems to me that public attitudes and the attitudes of our overlords can change on a dime. The public's attitude changes quickly during times of duress when the man or woman on the street suddenly finds themselves affected by something. Until then, they are often fairly docile. Put them in positions of sudden severe economic, cultural or racial stress and see the difference. The potential hostility in people can make itself known even when people just detect there's a mild threat hanging over them. I've talked with a few people about the freedom of speech issue versus the offense issue and the smart ones can see the potential slippery slope there and are getting worried. Others can see the problems they might have with migrants who have different cultural values. Some people find the way people are now trying to explain "what transness is" versus how it was explained twenty years ago so convoluted and patronizing it's offputting to the public, etc.
So we have some obstacles to overcome in the future. I don't see any of it looking easy. It could be made easier though by trying to keep dialogue open, speech free, and by asking people be treated equally, not specially.
