"I would like to order."
It is quite confusing how 想 and 要 are used sometimes interchangably sometimes not. I may not have grasped the subtilities but it seems to me that the allowed translations are not entirely consistent. Note that I do not use the simplified word-list input but free text, and so really need to understand what words to use as opposed to eliminating obviously wrong alternatives.
I think a helpful way would be to stick with patterns first
想 + V
想 + 要 + N
Hmm, it seems like 想要 can be used with verbs too, at least based on https://www.chineseboost.com/grammar/xiang3-yao4-xiang3yao4-difference/
Is it valid to say 我想要点菜?
It is valid. @NasuSamaruk0 has explained it. My advice was only for a more straightforward way to remember as a start, so as not to become too confused. (So I said "first")
For a simple statement like 我想要点菜, the difference is minimal and I would even think it's kind of redundant. Adding 要 reinforces the desire and determination of the subject on such matter.
If you want to speak fluent Chinese, you must start working hard now.
Here, 想 refers to "to wish/to desire/would like", whereas 要 means "to want".
想 is used when you wish to do/get something
要 is used to indicate what you will plan to do/get something in the future
This is made extra worse because the "We would like to order" question requires you to use 要. But this question, "I would like to order" requires 想. I think it's just broken questions. Report report report.
Do I understand correctly that the characters in the "order" word mean roughly "a bit of food/cuisine" when taken literally?