There are possibly two things going wrong here.
1) Make sure the tools you've used inside the Model exist at 10.3 and have the EXACT SAME PARAMETERS as they do in 10.7 vs 10.3. You'll need to use the different versions of the help to confirm this. (You can toggle the versions at the top). If they match (ie 10.7 does not have a new or updated parameter) you should be good. If they don't match, I'm not 100% certain that the down-save will fix the tools to the older version. If you determine this is the problem, you'll need to "fix" them in 10.3 and save them from that version.
2) After you do the simple Save-As and select the older version: DO NOT TOUCH the TBX with ArcMap 10.7 again. IE. Do not "Test" to see that it works. Depending on how you're "Testing" you might actually be upgrading the TBX back to 10.7. Simply save it, and don't use the new version again - test it with the old version.