Every hotel and front desk that are processing upgrades are different, so it can be difficult to form standard advice as to what works best. Personally I don't use online check in if I've not been upgraded and the app/website shows suites are still available. My assumption is that some agents might view those who have already checked in online as being "fine" with the room that shows as assigned, and potentially not offer additional upgrades proactively. This might not actually be how upgrades are commonly processed, but it could help you in some cases and that's good enough for me.

If I do get a suite upgrade, I check in as soon as possible since it seems less likely they might downgrade you if they think you've not seen the upgrade yet, and they have some reason they might want to perform a downgrade.