Reading around on here it sounds like a lot of people leave their inspection tray out unless treating for varroa...but then I also understand beehives used to have solid floors which would suggest there’s no harm in leaving it in either.
I would have assumed it should be left in over the winter to help retain heat...but again reading around apparently that’s not the case
You're right - as usual it's just one of those things that no two beekeepers are likely to agree on. When choosing a home bees apparently tend to prefer a solid sealed floor, which makes sense as they're looking for security and a safely sealed up home. Having said that, all of my bait boxes have at least partially meshed floors, and still caught swarms. As soon as they've settled in, and especially once they've got brood in the box to look after, the bees really don't seem fussed.
An OMF is more for us beekeepers to monitor what the hive is up to. Especially, over winter, or in early spring it's nice to have the board in so that you can take a look every few days and get an idea from the debris of how big the cluster is, where in the box it is, whether they're uncapping stores, or how much brood they might be raising - all without cracking the lid and disturbing them.
You can get a *very rough* idea of mite levels by counting how many dead mites fall through onto the board over a few days. Or better yet, you can do an accelerated mite drop (such as a single OA vape) to more accurately count varroa levels by seeing how many phoretic mites had been in the hive and killed off by the treatment.
When it comes to providing ventilation, or keeping in heat, that's probably even more contentious. I think it really depends on where your hive is located. If you're in a particularly cold, or very windy and exposed area then your bees might benefit from the board being in over winter, but personally I haven't found it essential at all. My default position is to leave the board out unless I'm monitoring something, or need the hive sealed for an OA vape.