Another strategy is to introduce the melody as part of the performance. In shape note traditions, this takes the form of singing the song the first time using shape names/solfege before starting the lyrics (which gives everyone a chance to think about the music without scrambling for words at the same time). In a more traditional church context it's common to have a musician play at least some of the melody of a hymn before voices join in. Seems best for moderately complicated but mostly repetitive pieces, and you do need to spend a little bit more time on each song.
Singing together in groups can be a great feeling, building a sense of togetherness and shared purpose. While widespread literacy means getting everyone singing the same words isn't too hard, how do you get everyone on the same melody? This has often been a problem for our secular solstices, but is also one many groups have handled in a range of ways. Here are the options I know about:
Use broader cultural knowledge:
Choose songs that are already well-known. A random group of people in the US will have maybe a few hundred songs they could get through well with no prep, that people learned from hearing them over and over. Some are children's songs (Old MacDonald, Ba Ba Black Sheep), others are well known older pop songs (Hey Jude, YMCA), holiday songs (Jingle Bells, Rudolph), folk songs (This Land Is Your Land, Amazing Grace), movie songs (A Spoonful of Sugar, Over the Rainbow), etc.
Write new words to well-known songs. At our gatherings we've sung songs adapting the music from The Mary Ellen Carter, The Twelve Days of Christmas, Why Does the Sun Shine, Sinner Man, etc.
Use written music. Many churches traditionally took this approach, some using shaped notes to be easier to learn, and it can expand to sight-reading four-part harmony for a very full sound. This does require more advance work: it's not enough to have a song leader and accompanist, you also need to find, buy, or draft an arrangement in appropriate notation. This also only works within a culture of singing from written music, or if your group is a big enough deal in participants lives (ex: weekly gatherings) that many will learn to read music specifically for your events.
Build up your own songs. If you keep doing the same songs with the same people, after 2-5 repetitions the group will know them. No one knows Brighter than Today outside the secular solstice context, but since we do it every year (and some of our attendees have heard it at events elsewhere) it goes well. This works a lot better with groups that meet more often: weekly is great; yearly is hard.
Send out recordings in advance. If people listen to recordings in advance they can show up with the melody learned, ready to sing together. Many people will only need to listen once or twice before they can join in with others singing as a group. This also requires more work from organizers, though, and attendees are often not interested in listening through.
Performances. Expect that most people in the group won't sing along, and a few people who already know the song or are especially good at picking it up do join in.
Call and response. The leader sings a line, the group sings it back with the same melody (ex: Chasing Patterns in the Sky). Unlike the others here, this doesn't even depend on literacy or a method of getting words in front of people. But it also really restricts what you can do musically, since most songs aren't a good fit for this format.
Easy songs. Some melodies are much easier to pick up than others. The more the melody does the obvious thing, avoids jumps, and is repetitive, the more a group of people paying attention can pick it up during the song. This is part of the approach of praise music.
Verse and chorus. The verse is sung as a performance, the chorus is easy and people pick it up. This lets you do something more musically interesting than if the whole song had to be this easy. If you're doing this you probably don't want to put up lyrics for the verse, or people will try to sing it too.
Visual guidance. A leader can use the height of their hand above the floor to roughly indicate the pitch of the next note, or the words can be accompanied by indications of the melodic contours. The imprecision means it's more of a hint than exactly communicating the melody, but because it's intuitive it doesn't depend on your attendees having learned a system for communicating melody.
Muddle through. Sometimes you just really want to sing something new and difficult collectively. It won't sound great, but that's not the point.
These can also combine: if you have a song that some people know because they listened in advance, others because they heard it last time, and others because they can read the written music, that could cover 60% of the crowd, even if none of those could individually. And trying to pick something up while singing along with a group where 60% already know it is much easier than one where only the leader is communicating the melody.
A nice illustration here is the evolution of Somebody Will at our gatherings. It is absolutely not an easy song: it has a wide range, makes some large jumps, isn't all that intuitive, changes keys, and has so many sections that I've color coded them on the musician slides I use. The first time we did it I think it sounded really rough. The second time we tried doing it as a performance, but we got a lot of feedback that for this specific song, which is thematically about participation, people really wanted to be singing along. But through sending out recordings in advance to some people, and then by repeating it often enough that a lot of people have picked it up, we now have it in to an ok place.
I was pretty sure I already wrote this, but when I wanted to send a link to someone I couldn't find it. If you do remember seeing this before send me a link? I'd be curious to compare!