By 'modern worship' - I assume you mean contempory christian music (CCM)
Okay - so:
...I'm still expecting musicians to serve this song and not their own interests.
Absolutely.
...learn the parts as recorded and then go from there...
Very important to know the original, get the feel down.
From there, we can safely branch out while still serving the song and working with bandmates.
I wouldn't say 'parts as recorded'. Consider that there can be multiple parts to learn as recorded (not just your own instrument). This can be a silly amount of work for volunteers who have day jobs. Full-time, different story.
Also, songs can have been covered in multiple ways by multiple groups.
So, I agree on some main thrusts here...
However, I want to offer a 'but'.
Consider one extreme. Should we play the original song exactly as recorded?
Definitely not. It will be at best a copy. At that point, just play an original recording.
Never mind that many CCM songs are more than a full band.
I would also say, anyone who would only play an exact copy, when not required to, is creatively bankrupt.
It is completely possible to break the mould and still serve the song.
On the other side (but not other extreme), if the whole band is comfortable with playing freely, sans click and track, it can give the lead flexibility to change things up in a dynamic way, the rest of us are able to follow impromptu.
Additionally, you're able to more easily work with lower skill players, as the experienced players can pick up some slack as newbies essentially learn to jam along. This lets a worship team double as a training platform. This is not possible if you're trying to replicate a specific version along with multitracks.
Strictly speaking, genuine worship doesn't need any instruments at all.
This also means there are many possibilities aside from copying some specific version of a song.
Regarding overplay:
Good musicians know when and how to 'overplay' in a given context, there's nothing wrong with it in P&W (implied: if it's not working, they're not good).
Sometimes discussions of not 'showing off' or 'overplaying' seem to veer towards, "We can't have good musicians!", which is laughable.
Honestly, CCM can be incredibly boring. Maybe because of the derivative/copying nature of it.
The music itself might be a part of the problem here.
Maybe the songs have driven people towards contempt.
There are additional comments and complications, and I have veered away from the original topic.
I'll end with this humble opinion (keep in mind that I assume musicians who serve the song and are not interested in showmanship):
Worship musicians should be genuine, you're not genuine if you're copying some 'original'.
I, for one, appreciate even an honest but amateur rendition over 'expert' rote copying any day.
Why not have something at least a little different? Most non-musicians don't care if it's the same or not, as long as it works.