First this is a very complicated issue and I absolutely dare anyone to give you a definitive answer to the true origin of Taiji. There are a lot of people in China researching this and multiple Taiji family members with their own origin stories and many do not agree.
But it can depend on what you want to call Taiji.
The Chen family says they invented it and if you are talking about the Taiji we all do today it could be argued that this is true. But this is not to say that something was not existent before that was used to create Chen style Taiji.
However the historical root is a Taoist named Chang San-Feng. If you ask a Taoist this is true if you ask the Chen family this is a myth. Did Chang San-Feng exist, I tend to believe he did. Did he invent the Taiji we all do today I tend to believe he didn’t. Did he have something to do with it? I tend to believe he did. I tend to agree with some of the Chinese historians (not all) that are trying to figure this out. Chang San-Feng likely came up with something similar to the 13 postures, but there were more than 13. This was later combined with something called Taiji qigong add a dash of Shaolin and BANG you have Chen style Taiji which is the original Taiji. HOWEVER there are now some Zhaobao people disputing this or at least that Zhaobao comes from Chen, which for all intensive purposes it does.
Is Tai Chi originally a short form developed into a long form so that teachers could make more money?
Likely no.
Taiji, if you are speaking the Chen family was actually longer. Laojia Yilu was originally 108 forms and I believe now it is about 80. Actually the complete reverse of what you are asking is true. Many of the forms where shortened to gain more students and make more money. But this is actually more of a modern, post cultural revolution thing.
Is it originally a Taoist martial art that is watered down with superficial movements?
Maybe yes and likely no. Was it originally Taoist, maybe yes. Was it taken and watered down? Likely no.
I would not want to be the one telling anyone form the Chen family that there art is a watered down form of a Taoist art. That could result in a very painful answer.
Finally, is push hands originally a part of Tai Chi?
Good question, likely yes.
Almost ALL Chinese styles have 2 person forms that are similar to push hands so it is likely that it is something that was originally found in taiji.
Now if you want to ask about the Jian forms, straight sword forms, then no those appear to have been added later.
Also taiji is a very generic term there are multiple styles of taiji and the following is just the names of the style recognized by the PRC; Chen, Zhaobao, Yang, Wu, Hao, Sun. And now it is possible that He style will soon be recognized as well. But there are many other family styles of taiji.
One final thing;
Taiji is from a time when it was necessary to use martial arts to save your life. What was originally taught by many was very effective for that and not watered down because to do so would get you killed.