Since this is about new legislation, the important point is (imo) what the laws & definitions should be, not what they are.
In any case, IDK if these reasons do or should preclude uber drivers' definition as employees. Flexible hours and/or having your own equipment isn't the norm, but it does exist within employment. So does piecemeal work.
Regardless, I think the situation merits new regulation. One reason is competitive fairness. Smartphone ubiquity enabled these companies to form business models that keep them just outside of the "taxi" & "employer" definitions. This isn't fair to taxis and employers and it may force others into also becoming non-employers, to avoid the burden of labour laws.
Less trivially, these platform tech companies are, in my opinion, monopolies. In fact, their operating rationale from day 1 is to eventually monopolise a large market. Uber raised so much money from investors, despite reliably losing money specifically because successful platforms are expected to become monopolies. Even if you define drivers as independent business people, they should be protected from monopsonist pricing (including terms) power. So should consumers and (very importantly in a monopoly situation) potential rivals.
Here also the existing definition of monopoly is not as important as what the definition should be. Current anti-trust laws were designed for the 20th century commodity and infrastructure monopolies which emerged at that time. New laws need to target the types of monopolies that readily emerge now.. that means platforms. Google's adwords platform & amazon's ecommerce platform, for example, are monopolies. They control enourmous market share. They give their own participant businesses an overwhelming advantage. They're both markets where network effects make market share an overwhelming advantage, locking them into place.
Monopolies are inefficient. The logic of antitrust is economic, not just moral.
I realize this is mostly a discussion about labour rights. These are ultimately important too. I do think the gig economy has advantages. It can't stay a loophole for sidestepping labour laws... but OTOH, I think diversity of employment types is important. Besides flexibility, these are low gatekeeper "jobs" and that has value. Regulation always carries a danger of uniformity. All that said, we can't just move on with the status quo, where Uber drivers have fewer legal rights because they use surge pricing instead of weekly shifts.