Except for Life Guard. I don't have ambitions but that shows up as a self-employed career at City Hall.
Lifeguard comes with IP expansion pack, it's a career track and it's not necessarily "Self-Employed." It's similar to the Day Care option you can get if you have Generations. You don't have to have Ambitions to have
those careers because they were built separately and not built on skills. If you wanted to be a Self-Employed Writer, FisherSim, Gardener, etc... then you would need Ambitions to actually have a 10-level career in those skills specificially rather than your career tab saying your Sim is Unemployed.
Alchemy as a career can only be unlocked by Ambitions, it's not because it's a career that requires both expansions, but rather it's a skill that can make money and Ambitions unlocks Self-Employment for all Skills that have money-making abilities that don't already have a career for them (like turning Gardening into a career, or Fishing which normally on their own can make your Sim money, but they don't have individual careers keeping a counter of the money). Ambitions just adds a counter for how much money a skill makes you and grants you bonuses at different levels. That's self-employed. If that makes any sense...
Edit* Wow I typed all of that out and realized a much easier way to explain it. There are Rabbit Hole Careers, Professions and Self-Employment. Ambitions is the only EP that can unlock Self-Employed, however, it also had a few Professions introduced. Professions are basically any career that you're hands-on with your Sim. They usually have multiple requirements to level but they don't always have to be "the best" as they do for Rabbit holes. Rabbit Holes, obviously, you can see and control very little. Self-Employed are skills that Ambitions built careers around, which is just a big money counter that will grant you levels and bonuses for how much money your sim made with that skill. Life Guard would be a profession, rather than Self Employed, whereas Alchemy is Self Employed. That's why you're able to become a lifeguard without having ambitions.