You should ask your supervisor for a letter.
Not having your supervisor write a letter is going to raise red flags. People are going to ask themselves, why isn't this candidate's thesis supervisor writing a letter? If you don't have a letter, you need to answer that question. Unless there's a very good reason to not have that letter, I think that it's better to avoid the question altogether. The reasons you've given so far do not seem good enough to me.
You suggested that (1) there may be more appropriate letter writers and that (2) your advisor might not be up-to-date on your current research.
I think these are not good enough reasons to not ask your thesis supervisor to write a letter and I think you can easily address these concerns:
You should also ask your postdoc supervisor to write you a letter. Not doing so will also raise red flags for similar reasons. If there are other great letter writers, ask them as well. For the tenure-track jobs I applied to, each expected three or more letters. There's room for everybody.
You can prioritize the more important or relevant letters. In your cover letter, it is normal to list the names of people writing you letters and it is normal to list those names in the order that you want them to be viewed. You can also specify in the letter which letters reflect more recent relationships, closer collaborations, or relationships closer to the core of your current research efforts.
It is not a critical problem if your advisor is not up to date on your current work. Presumably, their letter will talk the work that they know about and about your qualities as a researcher, colleague, and person that will not change. Also, you can point your advisor to or summarize the work you've done more recently.
I provided a copy of a draft cover letter, research statement, and teaching statement to each of my letter writers so that they could see how I was pitching and framing my work. I didn't do this because we were out of touch, but it seems like something similar would help address your concern.