This one is pretty difficult and requires a significant amount of emotional intelligence but we use a couple of tools including https://bunch.ai for culture analysis and then tools like https://www.qualtrics.com/lp/employee-engagement/

But nothing beats an old fashioned one on one with employee's.

I tend to use our own CultureView instrument that always gets to the root cause and provides a leading indication of the deeper dynamics of culture. Fast - insightful and focussed on stuff that works. www.5deepvitalsigns.com

I would keep it really simple by crafting a highly succinct survey question that auto emails, is set up in messenger or online portal as a button/sliding scale, where employees could access and respond easily daily (if they chose). Managed by IT HR and/or Legal, there would be one question that could be answered anytime or number of times, but requires each employee to respond, say, at least 5-8 different days per month (this gets us a representative sample of the working days in a year.) All unique identifiers, excluding name, would be coded for collection on the back end, for analysis, with a guarantee that no personally identifiable information will be used... (trust is important here).

The rough question would read. On a scale of 1-10 (cold/bad - hot/good - not so good/good) .. or some other relatable context) "Taking a deep breath in and slowly exhaling out, on a scale from 1-10, slide the arrow (fill the cup, select the number, color scale red yellow green gradient or some other fun creative visual) to the spot that best describes how your mood in this moment; 1 being not so good and 10 being extremely well". or something to that effect. Numerical measurement is important here, so which ever type of response option you give ensure it backs up to an ordinal ranking framework.

The data/insight from asking this one question can be immensely helpful discerning which leaders/leadership styles are/are not constructive to employee health at your organization.

1) the overall feeling that each team member individually has towards their contribution, and their satisfaction in working with the team;

2) The time that it take to go from acknowledgement that a problem or issue has arisen to devising an action plan resolving the issue; If this is typically a relatively quick process, that's an indicator that the team works well together. If it takes a long time to get from problem to action plan, that means there's alot of blaming and finger pointing, and lack of communication.

3) How successful is the team at executing their it's mission.

4) how much turnover is there on the team.

In my view, 360 degree feedback, done by a respected 3rd party interviewer, from all team members, done confidentially, can provide a valuable pulse on how a team is actually working together.