(mostly, less)

It looks like you have mostly invalidated the second sentence of your first paragraph, and rightly so.

For your second paragraph, following the link gets text that includes:

Google must be given attribution using the appropriate text or brand elements when Google results are shown.

So your second link is clearly meant for a usage other than the original use case (since the original use has Google showing the results itself).

So I'm not convinced that you have yet demonstrated that google has any policy against the practice orignally described.

If not illegal

At least you didn't say that it might be "stealing" or "a crime". But this smacks of the escalation of language that has been pushed by the I.P. industry (such as the RIAA). There are no laws that specify whether you can create a form that submits a google search. So using "illegal" ("contrary to or forbidden by law, especially criminal law", emphasis preserved not added) is a sloppy escalation that I find is often done as a ploy of rhetoric. I'm not saying that you consciously used that ploy, I'm encouraging people to consciously avoid emulating that ploy which has become so common that having or even observing an intelligent conversation about "intellectual property" has become quite rare.

Just say that you think something might be against Google's policy (or "terms of service" etc.).

As for the specific practice, I don't have any evidence at hand to show that Google has explicitly granted it. But I have made such forms and will do so again (I believe more than one such form can be found on this very site). If the target were not google, then I would have more reluctance to do such. And if google objects to such, then they can very easily prevent it. But I am confident enough in this specific case that I don't even find it warranted to invest effort to look for a statement from Google regarding the practice.

If you come up with a Google document that actually addresses the described practice (not just some other practice that might have some similarities), then that would be appreciated information. And your broader point that websites are within their rights to set limitations on how others use their site and that failing to heed those limitations / policies can have consequences is very valid.

(Updated as marked. My apologies as without that addition that point was not close to as I intended.)

tye raises an important issue. BUT no "++" because I believe some of the conclusions (and even some of the inferences) lack rigor or cogency (in the sense of "persuasive relevance.") To address a few of those:

(mostly, less)
It looks like you have mostly invalidated your first paragraph, and rightly so.

Hardly!

(update in response to tye's update above: even limiting the objection to the second sentence seems to me debatable because...)

Were Google to construe (note the use of the subjunctive here, too) the use as illegal and pursue its position through litigation, mounting a defense -- even if the claimant's case is baseless -- could be expensive to the defendant. Did I really need to be explicit about that?

So using "illegal" ("contrary to or forbidden by law, especially criminal law", emphasis preserved not added) is a sloppy escalation

OP used the word "legal" in the question. I don't agree that using the variant "illegal" is an escalation. In fairness, however, tye did make the point in a message, that OP did not ask

"is it legal(?)"

so this is a point upon which disagreement is more reasonable than I thought upon first reading.

If not illegal At least you didn't say that it might be "stealing" or "a crime".

No, I certainly didn't, nor did I intend to suggest either of those. Rather, I hoped that my opinion (OP specifically asks for "opinion") and words would suggest some points to explore.

In fact, I'd be very reluctant to disagree with the suggestion that "e;...against Google's policy (or 'terms of service'..." might be a preferred usage in other (or even most) circumstances...

...but (largely because IANAL) I'm also reluctant to agree with the implicit distinction between "terms of service" and an enforceable assertion of ownership rights. In January 2008, SCOTUS blessed the EULA printed on seed packets in a decision read by some experts as granting maximal credence to "shrink wrap licensing." In other words, some believe the highest court in the US at least leaned to the proposition that a seller can hold that opening the shrink wrap represents contractual agreement to license terms even though those terms can't be read without opening the package.

Your view seems to be more-or-less in accord with that of techdirt, which comments: "The real problem here is yet another artifact of bizarre intellectual property laws. .... Intellectual property laws try to make digital goods more like tangible goods, but due to the nature of those goods, it actually provides even more control -- such as when it comes to limiting what a buyer can do after they've bought the product." (yes, I see this addresses "bought" but could be generalized, I think, to "used the product").

And -- I hope this is no surprise, despite what seems to be your interpretation of my comment -- I agree. However, I submit that your advice to research Google's TOS for information about the specific practice would be better addressed to the OP, than to the author of a few cautionary viewpoints.