The issue is 100% repeatable for me. In my case, I found the problem when playing the Pilgrimage 1.94 scenario downloaded form Steam. I believe this is specific to Steam scenarios based on comments from T74993 and not specific to Pilgrimage, but I don't know for sure as I did not test with other scenarios. I can do that if needed.

1. Load ARMA and start playing a scenario. Save games if desired.

2. Cause ARMA 3 to exit in a bad way (simulate a crash or power outage so ARMA does not give you the chance to "Save and Exit".

3. When you restart ARMA and run the scenario, instead of getting a Resume button, you will get a Play button and it is as if your save games are gone.

Refer to T74993 for other notes about how this might be affected by default vs. alternate profiles. Though that issue was resolved, the underlying reason why it can happen still exists. The fact that if the continue.Arma3Save file is not recreated, ARMA 3 will not let you resume an old save game file.

The save games are stored in the following two locations for scenarios depending on which profile you are playing as:

Default Profile
C:\Users\(username)\Documents\Arma 3\Saved\steam\Scenario_Name.Map_Name

Alternate Profiles
C:\Users\(username)\Documents\Arma 3 - Other Profiles\(ARMA Profile Name)\Saved\steam\Scenario_Name.Map_Name

There is a Workaround as explained in T74993:

The work around is to copy your last save game file and rename it to "continue.ArmaSave", and then restart ARMA 3, you should now be able to Resume your scenario and access your previous saved games as well. This work around was explained by goertzenator on Sep 11 2014, 8:05 PM for Ticket T74993.