These sound like team-level constructs (what Stapleton et al., 2016, called "shared constructs"), so hopefully the individuals responded to items that were worded that way. In this context, I would consider team member as "raters" of team-level constructs, and you will see me refer to interrater reliability (IRR) below.
- You should not "ignore nesting" (e.g., by running a person-level model with cluster-robust SEs and tests) because the constructs are characteristics of the team, not of individuals.
- In principle, you could calculate cluster aggregates (team means) from individual responses, but those are likely to contain a LOT of measurement error. It is usually not easy to measure cluster-level constructs with high reliability.
I would run a multilevel CFA and check the assumption of measurement invariance (see Jak & Jorgensen, 2017, for details), and follow Lai (2021, p. 95, Eq. 17) to estimate reliability of team composites calculated by average individual scores across both items AND group members (i.e., simultaneously quantifying IRR and scale reliability).
You may notice Stapleton et al. (2016, 2019, also referenced by Lai, 2021, in his Eq. 20) advocating for a CFA with saturated within-level structure to model shared constructs, but I disagree with this (Jak et al., 2021). However, Lai's (2021) Eq. 20 could still be interpreted similar to Eq. 17, and Eq. 20 would be more robust to any misspecification of the Level-1 model.
Jak, S., & Jorgensen, T. D. (2017). Relating measurement invariance, cross-level invariance, and multilevel reliability. Frontiers in Psychology, 8(1640). https://doi.org/10.3389/fpsyg.2017.01640
Jak, S., Jorgensen, T. D., & Rosseel, Y. (2021). Evaluating Cluster-Level Factor Models with
lavaan and Mplus. Psych, 3(2), 134–152. https://doi.org/10.3390/psych3020012
Lai, M. H. (2021). Composite reliability of multilevel data: It’s about observed scores and construct meanings. Psychological Methods, 26(1), 90–102. https://doi.org/10.1037/met0000287
Stapleton, L. M., & Johnson, T. L. (2019). Models to examine the validity of cluster-level factor structure using individual-level data. Advances in Methods and Practices in Psychological Science, 2(3), 312–329. https://doi.org/10.1177/2515245919855039
Stapleton, L. M., Yang, J. S., & Hancock, G. R. (2016). Construct meaning in multilevel settings. Journal of Educational and Behavioral Statistics, 41(5), 481–520. https://doi.org/10.3102/1076998616646200