This answer is late but may still help someone looking to do a time series analysis. Simply, no, you cannot compare level 2 Landsat data across satellites. This seems to be frequently done, but doing so will result in conclusions that are for all practical purposes meaningless. I tried doing this myself and got strange results that were not consistent with MODIS.
The only Landsat data product you can confidently use for time series analysis is ARD (analysis ready data). This data product is only available for the United States (as of 2022), but the USGS has announced plans to release a global ARD product sometime in the future.
The solutions are to either use a single Landsat satellite, to use the ARD product for analysis within the boundaries of the United States, or to prepare level-2 data for time series analysis. A final option is to use a global ARD Landsat dataset published by the University of Maryland, however, this dataset only goes back to 1997. https://glad.umd.edu/ard/glad-landsat-ard-tools
To prepare non-ARD Landsat imagery for time series analysis requires (at a minimum) applying a scaling factor to equate two satellites (e.g., Landsat 7 with Landsat 8), or doing some radiometric calibration to put all the satellites you're working with on the same scale. I don't think either is straightforward. As far as I can tell there is no publication that derives a scaling factor to equate more than two Landsat satellites at a time. To calibrate the data (if you're working on a region outside the United States), it may be useful to look at the methodology the USGS used to create the ARD product for the United States.