I have bathed cats of my own, at home, in the kitchen sink, wearing the heaviest and longest Playtex rubber gloves on the market. One would sit and suffer quietly, the other would fight and yell. Warm water, Johnson’s Baby Shampoo, then rinse, wrap cat in a big fluffy towel and hold in my lap to keep warm until dry (and keep claws out of my flesh!). They weren’t filthy oily matted messes, just unusually grubby kitties who had gotten gunk in their fur I didn’t want want getting into their digestive systems. Once, when I had an indoor-outdoor Persian, it was fleas.
I didn’t sedate them. Never occurred me to take them to a groomer. Traumatic as baths are to them, I would figure it’s less hard on them being in their own home tormented by their own person than among strangers in a strange place.
Cat articles I have read say fold a bath towel in the bottom of the sink to give them a non-slippery surface, and have a helper, someone with two free hands to hold the cat while you bathe it. Someone brave and quiet who won’t let go.