Herschel, deleting your Gmail account or disabling it will have the same effect. I recommend removing it completely with the "-" at the bottom of the accounts list. If you ever want to enable it again, you can. Gmail is an IMAP mail account, and as such, by deleting it in any email app, including Apple Mail, you'll have no effect on your email or labels on the Gmail server. You could theoretically have Gmail installed on dozens of IMAP email clients, but realistically having more than four or five of them open simultaneously would have an adverse effect and cause problems with Gmail.
Regarding your Gmail folders residing in Apple Mail, since Apple Mail is an IMAP email client, if Gmail is active in Apple Mail and you were to delete a folder, yes, the corresponding label in Gmail would be removed. You do not want to do that...you want to either remove (recommended) or disable the Gmail account in Apple Mail. This simply stops the Gmail account from being displayed in the Apple Mail interface...nothing will be deleted from Gmail itself.
Regarding your All Mail folder in Gmail (or in Mailplane or Kiwi...same Gmail interfaces exactly) you can not remove the All Mail folder. Google has something pretty special with All Mail, Herschel, which IMHO makes it the king of email (that along with it's unmatched spam filtering). Every message you own in Gmail resides in All Mail, but considering you have nearly infinite storage in Gmail, its not a problem. In fact, it's a great place to look for some message you want to find from ten years ago...or from yesterday. What happens is every message in Gmail in All Mail "and" another label (folder). The word label best describes it because you can have one single message in All Mail, yet apple as many labels as you want to it.
Using labels is explained here - https://support.google.com/mail/answer/118708?hl=en
All Mail is explained here - https://support.google.com/mail/answer/6581?hl=en
Archived Messages are explained here - https://support.google.com/mail/answer/6576?hl=en
So, in answer to the last sentence, it's not Mailplane or Kiwi that dictate the use of All Mail...it's Gmail itself, and that's all they are displaying, along with their other niceties to go along with the Gmail standard interface. Best way to compare, open Gmail in a browser, then look at it in Mailplane, and then Kiwi. Additionally, any setting changes you make in Gmail (or Mailplane or Kiwi) carry forth to the other apps. Change your default font in one, it's changed in the other two. Mailplane and Kiwi "are" Gmail...just put into a neat Mac desktop app, allowing either to afford you features not available in the Gmail app. Good example is multiple account, contacts, or calendars. In Mailplane you can have a tab for each Gmail account, accounts for Google Calendars, or Google Contacts. You would need many tabs open in a browser to support this and it would be slow. Both apps have comparisons between their features and the Gmail browser app on their respective websites. Mailplane is the most polished now, but also the most expensive. Kiwi is the new kid on the block, but look for some really cool features in the near future.