There are two separate issues here. Entering the water with air in your BC and over-weighting. I will only address the first.
Air in the BC. Unless you are doing a dive profile that requires a negative entry like a drift or free descent off then having air in your BC does several things.
1) In case you slip or otherwise hurt yourself and fall in the water you are positively buoyant. This gives you time, which in an emergency is always nice to have, to recognize the issue and come up with solutions. If rendered unconscious, are injured or are momentarily stunned as a result of the fall, then it is a lot easier to recover the victim if they are floating. A tossed grab line from the back of the boat to someone on the surface is preferable to sending in Rescue divers to follow the diminishing bubble trail downwards. That is if you actually do have such divers ready to go and they actually recognize there is an issue to begin with.
2) It gives you a chance after entry to check your gear (or the gear of your students / buddy) and make sure the entry did not dislodge or shake anything loose. I have seen people lose masks, flashlights (night dive), regs (pulled out of their mouths), DC's, compasses, dry gloves, cameras, webbing tear, SMB's and fins on entry. Yes, they should have secured it better. Yes, they need to get good quality gear with proper connectors. Yes, they need to learn proper entry techniques, but "stuff" happens... Descending without a mask, reg or both can be disconcerting to some and it immediately complicates a situation unnecessarily. Note: I said "descending" not plummeting. Even a 10 foot per minute descent rate puts 2" more water over your head each second. You can drown in 2" or 200' of water equally. The very nature of dive gear means that the further you descend the more negatively buoyant you are. Neoprene and air spaces compress with depth, they lose buoyancy and you go down faster.
3) You have time to meet up with and coordinate with your buddy prior to descent. If something does go wrong on entry you have someone nearby who can help and / or you can amend your dive plan accordingly before going down. A fifteen second conversation on the surface saves 5 minutes of slate writing and hand signing at depth. They can assist you with your issue if needed and if you need to abort then you are both in a safe position to do so.
4) You can release air as required and keep a safe descent rate. I normally don't have a problem clearing my ears, but on occasion I have had to make a slow descent clearing every few feet. That little bit of air in the BC helps.
Now I am sure there are those in the dive community who if they found themselves at the end of 50 foot unplanned descent mired in thick silt with air turned off, no mask, no reg, loose weight belt and unzipped dry suit they could solve the issues in a calm and precise manner. That level of perfection will always be the dream. Reality is somewhat different, however. Everyone reacts differently to stress. A twisted BC strap might be enough to drive someone off the deep end (pun not intended...).
When I step off the dock or roll off a boat with air in my BC, I am not doing it just because I was taught that. It is because I have considered the technique and decided air in the BC it is a prudent and safe thing to do. Anyone who straps a tank on their back needs to examine their own personal limits and take whatever precautions they feel are necessary, or not. If you want to jump in without air in the BC that is your choice and you will deal with any potential consequences if any.
On these forums we have access to incredible levels of experience from basic OW divers, to instructors to the tech folks. Talk to all of them, ask opinions and then make up your own mind based on your training, comfort level and personal limits. What works for me may not work for you and at the end of the day it is your finger on the BC dump valve as you go below the surface. You are responsible for your own safety...
Last point I wanted to make was that checking gear (yours and your buddies) is a continuous process before, during and after the dive. It does not stop before you step off the fantail.
Safe diving all...