You should have checked your contract before agreeing to pay for the notice period.
Assuming you're in India, the contracts usually have a condition that either party (you or the organization) can pay the amount equivalent to the notice period salary in lieu of actually serving (resignation) / allowing to serve (termination) the notice period. However, there is usually also a sub-clause (that many seems to miss to read) that in case of a resignation, you are allowed to pay instead of serving the required notice period only if the organization allows you to do so.
If you have got a contract that mentions that the buy-out (that's the term usually used for this scenario) is possible only if the employer allows / agrees with - then basically you do not have any options other than relying on the employer's decision thereof. All you can do is put a request to your manager / HR - but whether that will be approved, depends on their choice.
In this situation, it's best for you to keep the prospective employer notified of the scenario which is developed now. I don't know why you would commit a joining date before finalizing the last working date in the current organization - however all you can do now is request for an earliest release and keep the new employer informed about the developments.
Finally, to answer the question in your question:
Can an employer make me sit in the company after my resignation mail even though I don't want to serve my notice period and pay out instead?
Well, they cannot physically force you to come to work, but in case of a no-show during the notice period, they can take certain actions, from as simple as not paying you settlement amount (and withholding relieving letter, in context of India) to as severe as filing a breach of contract lawsuit.