Are home workouts as effective as the gym?
Whenever you do something new you have to start at the lowest level you can achieve. If it has to do with weights and you cannot even do it with body weight why do you have to do it at the gym? Let's say you can only do 1 push-up and you weigh 70 kg. So you can most probably do a 55 kg bench press once. So why the need to go to the gym just to start bench pressing 35 to 40 kg? Waste of time and money isn't it?
The main point you bring up is effectiveness. Let's compare the most popular compound exercises using weights and their equivalent body weight exercise. The squat movement is the same. So is the bench press and push-up. Only the deadlift has no body weight equivalent as far as movement is concerned. But don't forget that the leg part of the deadlift has it's equivalent in the body weight squat. The lower back action is taken care of by the hypertension exercise. So not much difference there. So all in all the big 3 has it's body weight equivalent. Biceps, triceps and isometrically used muscles can also be trained with body weight alone.
So from the above, home workouts are as effective as going to the gym. Some will argue that you need proper coaching to do the exercise well. Why do you need coaching for body weight exercises? Are you so lacking in intelligence to not surf the internet for the best method to perform a body weight exercise? Moreover there is no danger from falling weights, using too much weight, etc which requires coaching. These people just want your money.
The only time you should go to the gym is when you have reached your max development with body weight and want to progress further. Then, you NEED the gym for the equipment and instruction. NOT before.