[v24.2.x] [UX-33] rpk: migrate login + profile creation flow to the Public API.

[vbotbuildovich]

Backport #24877 to branch v24.2.x. This issue was created as a placeholder because the original PR's commit(s) could not be automatically cherry-picked.
Here are the commands to execute:

git checkout v24.2.x
git checkout -b manual-backport-24877-v24.2.x-152
git cherry-pick -x 219adb2dbf ecfbbc8d1f b142aa2f44

git push origin manual-backport-24877-v24.2.x-152
gh pr create \
  --title "[v24.2.x] [UX-33] rpk: migrate login + profile creation flow to the Public API." \
  --base "v24.2.x" \
  --label "kind/backport" \
  --head "manual-backport-24877-v24.2.x-152" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --milestone "v24.2.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/24877 "

JIRA Link: [CORE-9222](https://redpandadata.atlassian.net/browse/CORE-9222)