[FrankenMauser]

What type of silhouette shooting?
Positional?
Prone-supported?
Prone-unsupported?
Bench-rested?
Upside-down, hanging from a rope?

Setting a rifle up for supported shooting requires a different approach than setting up for positional shooting, which are both different than setting a rifle up for benchrest, etc.
Which means the starting point can be important, and not every rifle is readily suited to modifications for each application.

Overall, your 10/22 is probably a good starting point. It can be tweaked and reconfigured to suit pretty much every shooting sport.
If these matches have "long" range targets regularly (say, 125+ yd), I'd slap a 20 or 30 MoA scope base on there, so you still have room to dial in the scope.
For glass, I'd recommend the Athlon Heras (FFP models). It is a lot of scope for the money. And you'll probably need it. In rimfire shooting, you'll see windage and elevation adjustments at 75 yards that a .308 Win or 6.5 Creed shooter doesn't see until they're at 400+ yards.

You'll want to find good ammo and stick with it. The most popular 'budget' option is CCI Standard Velocity - a subsonic 40 gr RN. Subsonic, because normal HV ammo can destabilize when it goes trans-sonic, which happens very quickly with .22 LR. Here, at ~5,800 ft asl, CCI Mini-Mag or CCI Blazer ammo, 1,235 fps rated MV, goes trans-sonic at about 72-74 yards. But if you drop to sea level, it is more like 60-62 yards.

I shoot NRL22. Our club shoots 25-380 yd on the main firing line, and up to about 475 yd for field matches. My primary rifle is a little shy of 10 lb. Heavy for a .22, but fairly light for the sport. About 20% of the rifles on the line at our local matches will weigh between 16 and 29 lb. Because all of the shooting is rested/supported, and the heavier the rifle, the more stable it is. (Club stages sometimes being an exception for 1-2 shots to start a stage.)

My son shoots a 10/22 compact that weighed around 7 lb last week, with a Magpul X-22 stock (for easy rail attachment and wider fore-end), and an Athlon Neos 2-12x. But, we just slapped a huge monstrosity of a telescope on there - an Arken SH-4 (4-16x) - in conjunction with with a new Kidd barrel, and it is 8.6 lb now. Nearly too heavy for him, but still light enough that most shooters would feel unstable and unsteady.

Most of the mentality of "heavier = better" crosses over to other disciplines, as well, like ARA Bullseye and PRS22. (To prescribed limits for certain classes, of course.)

In contrast, I have spent a lot of time, recently, talking with competitors in Steel Challenge. There, they want short, light, and as minimalistic and simplistic as possible. Of course, speed is the key, so nearly everything is semi-auto.
10/22s in Steel Challenge often weigh about 3.5 lb, or less; being a receiver and trigger group, small chassis and light weight stock, and no fore-end (they grip the barrel). I know one competitor that doesn't even use sights or an optic. "90% of targets are so close that you don't need them. The other 10% get taken care of with practice and familiarity." If he does get presented with a 40+ yard target, he uses the high bore axis to benefit, by lining up the crown of the receiver with the tip of the barrel, and holding bottom of target until he figures out the correction, if needed.