There are four possibilities, here.
I've got a Canon EOS 4000D [T100] and regularly shoot photography for my wife's church. I use a Tamron SP 24-70mm F/2.8 Di VC USD G2 and a Sigma 50-200 f/4-5.6 dc os hsm.
You could upgrade your body or switch to Canon's newer EOS R system. The sensor upgrade might get you 1/2-1 stop or so on high ISO noise improvement. But probably wouldn't be worth it for the expense.
You can also upgrade your glass: that 50-200 is not well suited for indoor shooting; you could consider swapping out the 50-200/4.5-5.6 slow lens for a fast prime (e.g. an 85mm f/1.8 or 100 f/2) or go really expensive and get a 70-200/2.8. While thinner DoF is something you'd struggle with, you can grab 4x the light at f/2.8 than you can at f/5.6.
The problem I'm having is it's quite fast paced, lots of movement and generally low light. The 4000D has garbage noise at higher ISO, so anything over 800 is very noticeable in my experience, with 800 being acceptable.
You could post-process for noise or just live with it. And also examine how you're exposing in lower light. Noise isn't just based on the ISO setting; it can also depend on how much signal (how well exposed) your image is. Noise is always more prevalent in darker regions of the image (less signal), so underexposing at iso 800 might look noisier than exposing properly at iso 1600. A lot of newbs assume that the lower the ISO the less noise they'll get so they minimize the ISO setting and accidentally underexpose.
Shooting wide open at F2.8 often gives too shallow DOF and using the Sigma is often out of the question unless they're almost stationary.
And... you can also use flash.
I do use a flash but it can be temperamental to get right.
Experience and study may help with that and how to use TTL (or M) to your best advantage. I highly recommend going to Neil van Niekerk's Tangents website to learn the basics of on-camera bounce flash. But generally? You probably have to use higher ISO settings with flash to light up larger space so your bounced flash is more fill than main illumination.
This is just me, but you paid the big bucks for a big sensor so you could use the high ISO settings; might as well use them.
Not sure if it's because I'm inexperienced with using a flash but I struggle to bounce [off] the ceiling. The unit we're in used to be a fancy pushbike shop so they had black slats hanging down from the actual ceiling (also black)...
The ceiling may be problematic; it also may not be. And you can bounce off walls as well as ceilings. It's all about choosing the right angles anyhow.