Addressing globally to the matter of "he said, she said" things and who says the truth and who lied, as well as brushing the matter why not placing the account on withdrawal-only mode, if I may chime in based on a simple guess, it might be a case where both parties are actually telling the truth. It just... OP had a bit less patience and understanding on how self-exclusion works, while SB is simply sticking to their procedure of self-exclusion.
OP, do you mind to kindly confirm if by this part of the post, the first email from SB asking for your withdrawal address come about one day after you asked for permanent self exclusion?
[...]
Regarding the emails, when I sent them an email about self-exclusion, I got a response within 5 minutes. But when I asked about withdrawing my money, there was no response.
The first email from them came the next day, around midday, but I have a feeling that if I hadn’t started making noise about it, I would have waited much longer.
[...]
From what I understand of the procedure applied by many casinos upon self-exclusion, the first 24 hours is always the cool off period, this is where the casinos let players to reconsider their decision and/or determine if they need to go straight to perma-ban the players due to Responsible Gambling Act kicking in. After that 24 hours, come the email asking if the player want to have certain duration of exclusion or permanently, as well as the email to process the remaining fund, suppose perma-ban happened and there are still some fund left on the account.
I checked SB's self-exclusion page to see if they apply the same rule, and they indeed do:

So, if I may throw a simple guess here, suppose OP got email from SB the day after his exclusion, it's not because they deliberately delayed the response and/or they had to send it because OP made noises in many platforms, it's simply the procedure of 24-hours period. Thus, regardless of OP's thread or not, on this forum or elsewhere, the email asking for withdrawal address will still bound to be sent.
While for the matter of withdrawal-only mode, now, I am not completely sure about it as, unlike above where I had several experiences with self-exclusion cases and thus have some understanding on how things work, I don't have an understanding on what casinos must do and must not do following a responsible gambling act kicking in, like what is negotiable and what is a standard that must be obeyed, but I think it is safe to assume that withdrawal-only mode is not possible because well, it's self-exclusion. It's in the name. The player is excluded in entirety from the platform, their account entered a full lockdown, no other option.
Plus, I think having withdrawal-only mode will kinda defeat the purpose of self-exclusion for gambling addiction as it has a potential of "taunting" the player to breach the exclusion [ban evasion] by multi-accing and/or to create an account in other platform by allowing them even the barest access to the platform. While not an apple-to-apple, a comparison will be to allow an alcoholic walk across the liquor isle to get the ice cream tub at the end of the hall because all of the bottles are locked and inaccessible to the person.
Thus, the safest way is complete lock down, and they'll hand over the fund without requiring the problematic gambler to access through the platform themselves, as the goal is to keep the problematic gambler as far away as possible from the platform.
I hope this make sense.
And OP, I also hope I can get you some understanding regarding the email situation. Regardless, you'll still get your fund by tomorrow. I'll see to it that you'll get them. Though I personally don't have even the slightest doubt that SB will send that fund without the need of a third party ushering them to.