To make more emphasis on your respective points and inputs here, i will suggest that if we are investing using an emergency fund, then that fund does not necessarily have to be in fiat, we can as well develop a means of creating a non custodial wallet and have our emergency funds kept on such, because the use of banks could be challenging to an extent that when they placed an embargo on an account, it may not be able to perform any transaction until they are done or settled with why they are doing so.
I believe that JJG have already addressed this issue of keeping your emergency funds in bitcoin. If you keep your emergency funds in bitcoin, for instance your three months salary is $1500 that you have used to set up your emergency funds. What if a real emergency occurs and bitcoin price has dipped with 30%, you will be at loss. This is one of the big disadvantages of keeping your emergency funds in bitcoin.
Secondly, since you said that your emergency funds should be in bitcoin kept in your noncustodial wallet. What if when you are hit with a real emergency and transaction fee is very high due to congestion in the blockchain. Fiat is the best form to keep your bitcoin investment, if you are complaining about your bank, you can keep the fiat in cash in your safe at home. It's good that when we are investing in bitcoin we save ourselves every possible means of running at loss during our accumulation journey
There is no doubt that keeping Bitcoin in your emergency fund will be a bad decision. From my point of view, emergency fund money should be kept in a place where you can use it at any time for an emergency. In this case, I would advise against keeping emergency fund money in the bank. Because all types of bank operations are closed due to bank closures on holidays, which is why it will not be possible to withdraw emergency fund money from the bank on holidays. Even online transactions or ATM withdrawals are not possible at times due to network problems. So, keep your emergency fund money with you as much as possible and keep it in fiat. So that you do not have any problem in using the money in an emergency and do not have to suffer from delays.
In this case, keeping emergency fund money in Bitcoin is by no means the right decision, rather it is one of the biggest wrong decisions. For a real example of why this would be a bad decision, you can keep some necessary money in Bitcoin for a few days that you will need after 15-20 days. If you are lucky, you may not get the results of this experiment right, but if you are not lucky, you will get real evidence why emergency funds should not be built with Bitcoin. You may think that your emergency moment is far away and you will benefit from it by then. But you should remember, the emergency moment is very close and can happen at any time and the Bitcoin market is very volatile.