I have a big fig tree in my backyard. I have always been removing all the fallen leaves this time of the year as I was told that any dead leaves could give rise to increase in insects and plant diseases. I wonder if this is true. I think that the dead leaves will eventually rot and put nutrients back into the soil. I should also mention that I have bark mulch ground cover which may slow down disintegration of the leaves.
Yes, you are correct that the fig leaves will eventually rot and the nutrients from them will make their way into the soil.
You should monitor the health of your fig tree throughout the year and look for any disease and/or insect issues. If you see fungal spotting on leaves then you should rake up the leaves and dispose of them such as in your yard waste container. Fungal spores from infected leaves can remain over winter and then when new leaves come out in the spring infect the leaves once more. By leaving diseased leaves on the ground under the tree over the winter you are allowing disease pathogens to build up, readily infect your tree the following year and increasing the disease load over time.
A thick leaf mat under a tree can impact grass by preventing light to reach those plants. This could lead to sparse grass the following spring and invite invasion of weeds and moss.
Additionally, one of the decomposers that love dead plant materials are slugs. Slugs use dead plant residues to hide and it also provides food for them. Some insects that attack plants could also be benefited by having a place to hide and overwinter under or within a think leaf pile, such as earwigs.
If you have been not picking up the leaves and have not had any disease or insect issues you can continue your current practice. If you have a disease or insect outbreak down the road you may want to rethink your fall/winter clean up routine.