I'm pretty sure that the main speaker's HF drivers aren't in very good shape because I had to boost many of the high and upper mids to get decent sound.
How much boost and how many bands? You should generally be cutting more than you boost.
I don't think it is possible to definitively identify the speakers from the very limited information provided, but if they are 25 or so years old and have not been serviced then they very well may have reached the end of their life without some repairs. However, if they are a two way box with a 15" woofer then it would usually be pretty obvious if a HF driver or crossover was toast and that would typically not be something you could correct with EQ. And while it sound unlikely, you may want to make sure you don't have something like a biamped system where all you really need to do is turn up the HF amplifier or turn down the LF amp or any other processing in the system.
As far as repairing or replacing the speakers, it they are more than a few years old then it is pretty common for a church's needs and expectations related to audio to have changed. So you might need to assess not only if the speakers are performing properly but also if they are capable of serving the church's current and near future needs and goals. If they can provide acceptable output, response, coverage, intelligibility, etc. then at least looking at repairs may make sense, otherwise it may be more effective to look at replacing them.
I'm not concerned about people "messing with" the EQ settings on the mixer so whether the processing is separate or not isn't much of a concern. The person who primarily runs the system is pretty overwhelmed with the whole digital mixer thing so I doubt he'll ever be doing much more that basic mixing(pushing the faders/mute buttons and minor channel EQ settings)
Why would a church purchase a mixer that their primary operator cannot operate effectively? And if there are overriding reasons to do so, wouldn't it then be a priority to help that person become comfortable with operating the mixer?
I suggest you save whatever you come up with to multiple presets on the mixer since it sounds like the settings and even the base preset scene could be inadvertently changed or overwritten.