I have run multiple differing games in order to try and figure out the issue behind not being able to immediately registered at City Hall as Self Employed - Band career. I've run it by having a solo sim household and a full band member household and this is what I discovered for those still consistently having issues with its function:

1) Since many still have questions about how the process is supposed to function, I will say, YES, you should be able to IMMEDIATELY apply as self-employed under the Band career without any mods whatsoever as soon as you ask even one additional member to be in your band. You won't get any gigs until you've completed the round out of all 4 members though.

2) The Nuts and Bolts of it: When you start JAMMING with other Sims and increase even a single point in ANY instrument, you'll likely get a Happiness Wish to form a Band. When you ask someone to join your band you get the Memory "Master of the Band". This puts the "Band" skill into your Skill Journal. For the Band skill it is not 1 - 10 but -1 - 1 [i.e -1 you're not qualified to start a band; 0 you're qualified to start a band because you played an instrument to level 1 & jammed with other Sims; & 1 you've created a band]. You either have it 'Mastered' -- meaning you've formed a Band, or you do not. SO! As soon as your Band is formed and named the Skill system should have changed the Band skill from 0 to 1. As soon as that happens, you are 'high enough skilled' to be registered as self employed.

How I confirmed this is through the Memory Scrapbook with a 4 member household with 1 Sim unemployed and 3 employed in other careers. When I initially asked a Sim to join my Band, we both get the memory icon over our heads, and the message stating we will always remember becoming "Master of the Band" then the picture of us shows up in our scrapbook. Each successive Sim I add to round out the 4 member band gets the same "Master of the Band" memory. Now when I go into MC for each of the 4 band member Sims under MC>Advanced>Skill Level>Band it is STILL set as 0. This shouldn't be. It should have switched to 1. So, I manually switch it to 1 for each member of the band... and they each get the pop up Memory "Master of the Band" a SECOND time after it is manually changed as well as a 2nd picture of it in their scrapbook. Thus, this confirmed that the switch is not taking place as intended for whatever reason.

3) Why is it Not Functioning as Intended??: I have no idea why it's borked. Since I was able to form a Band when I was only playing up to ALL expansions from Base to University, I believe the borking came from the Opportunities issues that arose with the Into The Future expac. I have lots etc that use stuff from ITF as well as Islands now, so I don't want to be forced to go back to University as my base game just to be able to play Bands. While researching the issue with Band skill, I discovered the mod that fixes the Opportunities issues BUT that doesn't necessarily resolve the Band issue because those opportunities are pretty much hidden and, even after research, there's really no clear indication what triggers a specific opportunity for a gig and which member will get it. I can confirm that it isn't always the leader of the band or members who are self-employed as a band. It's very random and few and far between.

What I can say is that, if you level up your instrument skills past 5 and haven't gotten any gigs, you likely won't get any because it seems progression skills based. That's how I worked out the fact that the system wasn't switching the band skill from 0-1 like it should with the last 2 expacs. Just like with any other career, you going from level 1 to level 10 as Self Employed - Band is supposed to level up along with your instruments skills and the only way you can make money to level up is by getting GIGS, not merely jamming for cash. You can jam for cash as a band to EARN money but it won't apply to your Self Employed career advancement. THUS, if you don't switch over the Band Skill from 0-1 using MC, you won't get the agent, bar, or lounge calls for GIGS to advance in your career.

4) I have confirmed that when I make a band using the Opportunity mod fix and manually switch the Band Skill from 0-1 on each individual Sim in the band then register at City Hall whichever 1, 2, 3, or 4, it doesn't have to be the entire Band, as Self Employed - Band, they will get calls for gigs and earn money towards leveling up that career.

So, bottom line is that if you're using ITF expac and want to do a band, you will likely need to use MC as well as the Opportunities mod fix in order to have it function as intended. Lastly, I'll add that you should switch over your Sim's Band Skill from 0-1 immediately after creating a full member Band to ensure that you start getting the calls for gigs.