How do I compensate for lack of minimum shutter speed when in Av mode with the Canon 750D (Rebel T6i)?

I’m fairly new to photography, and I own a Canon Rebel T6i. Besides manual mode I’ve learned about Tv and Av modes, which to be honest I really liked, mainly when I’m shooting on events with changing lighting conditions and I’m find myself too slow to change the settings for every shot. I like the idea of the Av mode very much for event photography, however the last time that I tried I had a huge problem, the shutter speed was too slow (I was using a 24mm f2.8 pancake lens) so when I got home I find out that most pictures were taken at 1/25 sec, even some at 1/60 were blurry, given that I don’t have super steady hands and people were moving.

After some research I found out that some cameras allow you to set a minimum shutter speed when in Av mode. However my camera doesn’t seem to have that option. I would like to set the minimum shutter speed to at least 1/125 (1/200, would be the best for me right now), but I can’t. So I often find myself shooting events at Tv with a shutter speed of 1/125 or 1/200, but the results aren’t ideal since I can’t control the aperture.

What I’ve tried lately to try to compensate for the lack of a minimum shutter speed in Av mode is to set the camera to manual with ISO on auto. This way I can set the minimum shutter speed (but I’m stuck to it, if the lighting conditions improve I could be taking pictures at a higher shutter speed) and I have the control of the aperture, leaving the ISO to balance the exposure. Far from ideal, I would guess, however is there a better way to achieving this with my Rebel T6i?

I know I could just be using full manual, but as a beginner it can get pretty awkward to have people in events posing and waiting for the to find the best exposure in full manual. Is there a better way to do this or should I just embrace full manual.