I generally try to keep my campaigns fairly open-ended and avoid railroading my players, or keep it to a minimum when it is performed. Generally they always tend to have a plot hook or two that they can follow, and tend to be invested enough in the presented hooks that I don't have to worry about them going too far off.

However, I have noticed at times, some players tend to have what I would equate to a "Well what do we do now that were here?" look. While none of them have explicitly expressed that feeling, when talking to them about the sessions I sometimes feel like they were struggling with deciding how to approach some of the more open ended situations, especially after some time or adventures have passed between receiving their original goal, and having the ability to act upon it.

Is it considered appropriate, to, as a DM, tell a player the thoughts of their character? For example, after describing a town they arrive in, saying something like any of the following examples :

"Your mind races with thoughts of the upcoming tasks at hand, perhaps you should look into [Group X], or go talk to [Knowledgeable Guy Y] about [Bad Thing Z]."

"The town guard stand vigilant and alert at the gate, noting down those who pass through, perhaps they may have seen [FindThisGuy A]?"

I know some players can be sensitive about being told what their characters think, but the thoughts I'm proposing are strictly a question of knowledge and goals, and not opinions of what characters think of a particular person or situation.

How close to railroading is this, and is this even an appropriate technique for a GM to use?

Giving players their character's knowledge is acceptable.

tl;dr Provide description, even if it's realizations internal to the character that maximize player agency.

Provide knowledge based hooks

Class or domain knowledge that affects the story or provides a hook

"Being a trained fighter, you noticed that the guards at the gates were wearing ill fitting uniforms and were holding their spears awkwardly."

"The rogue in the group overhears some peculiar phrases, and recognizes them as theive's cant. He overhears that Mr. Y is a very knowledgeable guy."

Character Knowledge

If the players are formulating plans, it can be useful to include their backgrounds or other aspects to give them some insight into what may or may not work. E.g.:

"The bard spent his childhood as a beggar, and would know that approaching the urchin by forming a ring around him will only serve to drive him away."

Avoid directly dictating thoughts or intentions

This is clearly the demesne of the player. It can remove player agency, and make the game more of the GM running the entire show. The GM should narrate the stage and the reactions of the world to it. Not the reactions of the players to the world. That's the player's role.

Giving players clues as to how the environment is affecting them.

"The foppish noble insults the wizard's shabby clothing. It's clear he's attempting to anger the wizard."

This allows the player role play their wizard's reaction. Do they get angry and react? Do they play it cool? Do they shake it off while singing some bar song they learned last night? That's up to the player.

Generally, you should not tell your players what their characters think

The GM controls the entire world except for the player characters. What the player characters are thinking and how they are emotionally feeling is generally left for the player to decide.

Of course, there are some minor exceptions for when some effect, especially a supernatural one, causes a character to think or feel something specific, but that is not what you are asking about.

You can and should tell your players what their characters know.

The Characters will know quite a bit that the players will not know. The Players after all have only a distant and murky view of what the characters are experiencing much less what they learned through decades or even centuries of life in the foreign setting of the game world. It is perfectly appropriate to tell the player that their character knows certain things that might be relevant.

In fact, many knowledge skills are there precisely to provide a mechanical structure to that kind of knowledge. I would never hesitate to call for a check against an appropriate skill to see if the character knew something pertinent, or make it myself behind the screen.

I would never phrase this as "Perhaps you should check...". It is the Player's job to decide what they should do. But a more neutral statement of "Your knowledge of the city and its people tells you that X might know something relevant" or "You know this city keeps fastidious records and you know that Person Y has access to them" is helpful and appropriate, especially if you tie that knowledge in to a specific character that has reason to know it.

Other options

When the players seem to be at an impasse, and the plot is at a place to support it, remember you can also have information find them. For instance, if they are investigating a criminal organization, but run out of leads, they might stumble over a crime and learn in the process of foiling it that the criminal is a low-level member of the organization. While this could feel somewhat like Deus Ex if not handled well, a version of this is done frequently and sometimes openly be prosecutors in the real world where it is sometimes called ladder climbing.

Alternatively, if the characters have been loud around town about what their goals are, such as by asking in every tavern there is, someone with information may well approach them offering to help, for a price... While not precisely the same, in the real world many "mixers" or "networking events" are explicitly set up to help facilitate that kind of meeting.

Overall, it's a good way to deal with deadlock, but...

I try to avoid it. There will be players who are really rubbed the wrong way by this sort of thing, a "don't tell me how my character feels" kind of deal. I try to avoid those players when I'm DMing, but it doesn't always work out.

The solution I've come up with is to speak in a sort of third person about it, using "ing" verbs. For your example, I would instead say something like:

"The upcoming tasks at hand are enough to make anyone's mind reel; looking into [Group X], or go talking to [Knowledgeable Guy Y] about [Bad Thing Z] could prove useful."

As for this one, it seems like a perfect segue. It's borderline close to a video game exclamation point, it gets the job done.

"The town guards stand vigilant and alert at the gate, noting down those who pass through; perhaps they may have seen [FindThisGuy A]?"

A lot of groups don't need this, but most groups would rather get along with a little help from "above" than awkwardly stare at each other, floundering over their next course of action.