suggestion - option to exclude private profiles

Not possible.

1. Easy to bypass - just switch to public, join GA, set private again. Also would require many rerolls, and what if you request a reroll when it's private, when support checks he set back to public again?
2. Would generate massive server load - atm SG Sync account with steam once a week and then you have to set account to public. Other that that SG is not checking your Steam profile all the time. With your idea if you set this rule SG would have to sync account each time anyone was entering GA. 1000s of ckecks per GA - server would die, plus Valve would blacklist SG for DDoSing (as we would also generate a lot of requests on Valve servers);p
3. In case of Steam community servers crashing (happens a lot) or lagging suddenly your GAs become unaccessible by anyone because SG cannot check profile public settings. And then it generates more troubles - people spamming forums and support that they cannot enter GAs even if their profiles are public, people losing Value of their GAs because they didn't get 5 entries b4 GA ended, because noone was able to enter shortly after it started and for example only 1 person joined.
4. If you think that person may be breaking the rules hiding behind private profile - ask support in reroll request for checking him - they have access to Synced data, if he indeed broke the rules he will get suspended and you will get your reroll granted.

you can use sgtools + invite-only giveaways to avoid people with private profiles :3

Just explain that SG Tool can't do the usual automated checkups unless they briefly switch their profiles to public and that you would have to check them manually since "Friends only" privacy setting doesn't work with SG Tools.