I reported the question to support. Their final answer is:

"The absence of all of these buttons and controls related to animation or automatic progression of a Manipulate[] slider is intentional, as this functionality is not yet ready in the Wolfram Cloud"

To me, this explanation sounds a bit strange since the Animator-control works as one can see from above.

But I can live with it since I have the bypass mentioned (Animator-Control plus InputField-Control).

Hi Rohit, I know that it works with an Animator-control instead of a Manipulator-control. But there is no input field with Animator, hence you cannot set a particular a-value. (BTW.: The Appearance->"Open" is unnecessary with Animator).

So I try it with an additional control for a. This is a reasonable bypass for my problem:

Manipulate[Plot[Sin[x (1 + a x)], {x, 0, 6}], 
 Control[{{a, 1}, 0, 2, Animator, AnimationRunning -> False}], 
 Control[{{a, 1, ""}, InputField}]]

This works and when deployed it has the same controls.

CloudDeploy[
 Manipulate[Plot[Sin[x (1 + a x)], {x, 0, 6}], 
  Control[{{a, 1}, 0, 2, Animator, AnimationRunning -> False}], 
  Control[{{a, 1, ""}, InputField}]]]

CloudObject["https://www.wolframcloud.com/obj/90b72a4d-6efa-463c-99d4-\
f2222dc19e05"]

An even better solution might be a self-defined control with FormObject that defines an Animator with an input field. This is pretty tedious and I did not really try it.