That's absolutely correct. There's no need to fiddle around with a mouse to write something for it to be valid. Here's what the US law on electronic signatures has to say on this topic:
In the United States, the definition of what qualifies as an electronic signature is wide and is set out in the Uniform Electronic Transactions Act ("UETA") released by the National Conference of Commissioners on Uniform State Laws (NCCUSL) in 1999.[8]. It was influenced by ABA committee white papers and the uniform law promulgated by NCCUSL. Under UETA, the term means "an electronic sound, symbol, or process, attached to or logically associated with a record and executed or adopted by a person with the intent to sign the record."
But a question has to appear.
How can a court decides if there was an intent for electronically signing the document, in case the signer denies his/her intention?
When it comes to intentions, it is NOT easy to prove it!
I had worked with electronic signatures for years, but lawyers have generally chuckled over lay folks focus on signatures. As you rightly say intent is the only thing important.
However your argument about how a court decides if there was intent is also true for traditional hand written signatures.
In http://agree2.com we offer signatures, but only as part of a whole package from drafting, invitation, negotiation, signing and then management. The whole process is recorded, providing a clear audit trail should something ever reach courts.
More important as well than signatures are the contents of what you are signing. Most lawyers I know have never heard of a case where someone questioned a signature.