Short answer: politics killed the "driver's license medical" idea for private pilots, but there are still reasons why BasicMed has significant advantages for some pilots.
The literal answer to your question is "nothing": the third-class medical hasn't changed at all. What you're referring to is an additional option called BasicMed, which you can hold ...
Looking at section 2307 of H.R.636, which is the congressional bill which implemented BasicMed requirements, it is clear that the physician who performs the exam, and provides care to the subject is responsible for determining what drug interactions there might be, as well as the impact of medical conditions, and whether they might be disqualifying.
Short answer: the safety pilot with BasicMed must be acting as PIC, but all other regulations and considerations are the same (i.e. as if the safety pilot had a regular medical).
You have quite a lot of points in your question, but I'll try to break it down.
"[...] aren't safety pilots acting as PIC because they are required crewmembers?"
Not always. ...
From the FAA's FAQ:
Q35: Can my AME provide the medical examinations under BasicMed?
All AMEs are state-licensed physicians and may elect to conduct
physical examinations required under BasicMed. But the FAA does not
require AMEs to provide medical examinations under BasicMed. It is up
to the individual AMEs to decide whether to offer ...
FAR 61.113(i) authorizes flight as PIC when using BasicMed. Therefore, your buddy can act as safety pilot if he is also acting as PIC.
If you were acting as PIC, that would make your buddy SIC (as a required crew member), which still requires a traditional medical certificate.
Yes, you can use BasicMed in place of a third class medical. When BasicMed was introduced, the FAA updated 61.23 (I've bolded the new text):
(3) Must hold at least a third-class medical certificate—
(iii) When taking a practical test in an aircraft for a recreational
pilot, private pilot, commercial pilot, or airline transport pilot
I think you may be confusing Basic Med with No Medical At All, some operations do not require any sort of medical exam
(b)Operations not requiring a medical certificate. A person is not
required to hold a medical certificate -
(1) When exercising the privileges of a student pilot certificate
while seeking -
(i) A sport pilot certificate ...
I guess it depends on whether you intend to fly a larger aircraft at higher altitudes at faster speeds or one with a limited number of passengers at lower altitiudes and airspeeds. If the latter is going to be the only flying you do, then a basic medical will be a good choice. If you intende to undergo additional training and obtain additional certificates,...
I have Kaiser insurance and tried really hard to get them to conduct my physical, to no avail. I was thwarted by some degree by the fact that my regular physician is not seeing patients right now, but my suspicion is that she may not have done it either. However, I have also heard of Kaiser members who did convince their physicians, so apparently there is ...