When it comes to non-mainstream religions 2000 years ago, we have a big issue with sources. First, lack of sources being made, when the religion was quite small; in a population of millions and millions, finding sources made by a religion that in its first century was probably measurable in, at most, thousands, is pretty difficult. Add to that the legal issues they faced, and they were severely constrained.
The second, as with all ancient history, is preservation of sources. We know very little about the administration and politics of ancient Persia. Is that because they were illiterate and recorded nothing? No, it's because the materials they used to record and write have mostly just rotted away. We have no contemporary accounts of Alexander the Great, only later accounts after his death. Is this because no-one ever wrote about him, or his invasions of their homeland, while it was happening? That seems very unlikely. It is presumably because those sources have all rotted, been lost, been thrown away, been rubbed out because someone needed to do some writing and didn't have any spare paper, etc.
Now, luckily, there is an advantage when it comes to early Christians; they loved making sources and spreading them, and copying them. We've got, for example, a lot of works by one of the earliest and most significant theologians and missionaries (Paul), because he wrote a ton of letters, sent them to different places, and people in those places copied them and spread them to their friends. Later they were copied again and made into codexes, which was quite unusual for the time (most people preferred scrolls). As it happens, codexes require better quality paper, and have covers, so are less likely to get damp and rot. See u/White__Velvet 's excellent answer about scrolls and codexes here:https://www.reddit.com/r/AskHistorians/comments/e037o8/when_was_the_idea_of_a_single_christian_bible/
This does make the study of ancient Christianity much easier than the study of, say, the worship of Mithras; I have great admiration for Mithraists, because a)they have to work from a tiny number of sources, made by a deliberately secretive religion, and b)they have to wade through so many conspiracy theories.
Now, saying this... just because Christians were slightly better at producing and spreading sources than, say, the mystery cults (whose study is as difficult as you'd expect from the name!), they still faced the problems I already outlined. There are many, many early Christian works that we have only bits of. The Gospel of Thomas, for example, is so short it's only a couple of pages, but it's quite likely that it was actually longer, we've just only got bits of it. And there are many early Christian works where we definitely only have a page or two, or even part of a page, because the rest rotted. There are early Christian works we know existed, but have very little idea about, because they're completely lost and all we have is a mention in some other work.
And this is before we even get into the issue of fakes; while scholars are pretty certain we have some genuine works of Paul, for example, there's a pretty good chance that not all the works of Paul we have are really by the same Paul, or indeed anyone called Paul. Then you've got Gnostic gospels, Infancy gospels (basically 2nd century fan fiction), etc.
So, now we come to pictures. Pictures are much harder to produce than writing. Elite, rich Romans could make mosaics and statues, but early Christians were not generally very rich. Statues would have been way too obvious for a religion that was at times persecuted (while later Christians tended to talk about the brave early Christians who proclaimed Jesus and went to their deaths... during the times of persecution most Christians did all the Pagan rituals they were required to, then went home and practiced Christianity secretly).Now, they may have drawn pictures, but most people are not trained artists. If I draw a picture of a man, it's generally just going to look like... some generic man. Roman statuary was of incredible skill and precision, but Roman mosaics were, while beautiful, not very precise with faces. It's also much harder to copy a good picture than to copy writing, so if someone did draw a picture of Jesus, it'd be much harder to make a good copy of than it was to make a copy of, say, Paul's letter to the Church at Corinth. I mean, were I to go back in time to meet Jesus, I could write down stuff he said, but I could not draw him. I could draw, like, a tall stick man with a beard, or a short stocky clean-shaven man, but nothing more complex than that.
There's also the issue of paint. Did you know that many Roman (and Chinese) ancient statues, those pure white marble ones, were actually painted? The issue is that paint doesn't last well (though it depends on the type of paint). Any pictures of Jesus painted in the 1st century would have flaked away.
So, let's look at what early pictures we do have.
This is possibly the very earliest picture of Jesus. It seems to be anti-Christian grafitti mocking Jesus, and was found carved into a wall in Rome.
https://churchpop.com/wp-content/uploads/2015/03/41.jpg
The key here of course is that it's carving, so easily preserved.
Here are some from the 3rd century.
https://churchpop.com/wp-content/uploads/2015/03/15.jpg
https://churchpop.com/wp-content/uploads/2015/03/25.jpg
https://churchpop.com/wp-content/uploads/2015/03/33.jpg
That last one is from the wall of a 3rd century Syrian Church. It's quite possible that there were other early Churches with pictures, but in the 1st and 2nd century Christians weren't generally able to build large permanent structures suitable for painting, and if there were any others they've obviously faded or cracked away.
Notice how, well, general that Jesus is. These are not good quality pictures where you can see his facial features. All we can really tell is he's a clean-shaven man. That's it. And that itself is odd; most Jewish men of Jesus's time had beards (but not long hair, as he's often depicted today). So it's honestly likely that by this time, any actual pictures of Jesus, if they ever existed, had long since been replaced by a depiction of Jesus that looks like his worshippers; the short-haired clean-shaven Graeco-Romans who produced these pictures.
So, TLDR; early Christians likely found it easier to produce text than pictures, and certainly found it much easier to replicate text than pictures. It's quite possible they did produce some pictures of Jesus, but if so these have simply vanished into the mists of history.
I have one last point to make, but this is a bit more speculative.
For a long time, historians have believed that traditionally Jews never portrayed God in pictures (except perhaps for his hand), or often even the Prophets. It's therefore possible that Jewish Christians (for example, all the founders of Christianity, and most of the writers of the New Testament) did not have an artistic tradition of depicting religious figures, and may have found the idea of depicting the son of God to be idolatrous. Whereas Greek Christians, who came to eclipse Jewish Christians, had a long cultural tradition of depicting divine figures. As they became more dominant, they began to depict Jesus, but had no actual idea what he looked like.
The issue here however, is twofold. One is that there are no real records of this argument. The second is that in the last ten years, some scholars have questioned the idea of ancient Jewish aniconism, claiming this is a modern invention. So you should definitely take this last point with a pinch of salt.
