It all depends on who your authorities are.
I accept the statements in the Puranas themselves, and there is much authenticating evidence, with more being discovered or admitted every year.
According to the Puranas, Vedavyas wrote down on leaves (which were used as writing surface up until a few hundreds years ago) 5000 years ago at the advent of the Kali Age. So the Sanskrit written characters are 5000 years old. Much of what has been accepted as standard Indology was knowlingly fabricated by British historians and linguists in order to denigrate the country they invaded and plundered.
Statements from some of these early Indologists have been found wherein they admitted their purposeful lies about the great antiquity of the Vedas and archeological evidence.
Sanskrit is Indo-European, as is Latin, Greek, German, Tocharian, etc.
Much of the argument for claiming a more ancient origin for Sanskrit is based on a hijacking of Dravidian tradition.
BTW, Sumerian is considered to be a "close relative" of the Dravidian group, but not exactly a cognate of any extant Dravidian language. The Sa'ami languages are also more closely related to Sumerian than to the so-called Fenno-Ugric group (which still provides a lot of vocabulary to the Sa'ami languages but not the grammar).
Whatever the Brits thought is irrelevant. They have the same problem ~ they're stuck within the more primitive and less ancient Indo-European language group using borrowed alphabets.