Keto is bit more than meat and veggies. I've been on keto for years to control diabetes. Gluten free is easier, IMO. Keto requires reduction of total carbs to such a low amount daily that it makes the list of acceptable foods more limited than gluten free does. Also, gluten free does not require that you get the bulk of your calories from fats and oils, keto can't work without this.
Keto isn't everyone's cuppa. It is significantly more expensive, it is a hassle when eating out or at friends' homes, and
there is no such thing as close enough will do. Either you are in nutritional ketosis (you need a meter to find out), or you are not, and that is determined by whether you keep total carbs from
all sources down to roughly 30 grams/day or less, which is not easy. Really, there are only two reasons to do keto, IMO - to lose weight (by far the best tool for this), or to control type II diabetes. But it will not work for either unless you control the carbs every day, day after day, eating at home or eating out. Not many people can stick to it.