The issues you describe have existed with signatures from the beginning of their use. There exists a tradeoff between ease of use and reliability, both of forgery and of people falsely claiming forgery.
Originally, the closest thing to a signature was the use of seals and signet rings. While relatively hard to forge, it only showed that the possessor of the object agreed.
Signatures, especially in cursive font, were developed later. They were in some ways easier to forge(you didn't need to get access to a physical device), but more difficult in others(the seal symbols tended to be used on everything and various improvements in technology had been made), and harder to falsely claim forgery(because most people can't alter their handwriting well). You were affixing your name to the document, indicating that you agreed. Often, the signatures were required to backed up with the signatures of other people as witnesses. They didn't have to agree to the document, they just had to agree to testify that you signed of your own free will.
Because witnesses, especially trustworthy and independent witnesses, are hard to come by, some places have dropped that requirement, such as checks and signing a aper receipt when using a credit card. But for some important documents, certain jurisdictions still require witnesses, including large transactions (a document relating to a car insurance payout I recently had required a witness to confirm my signature) and marriages.
However, with electronic media, the point of a signature is more to indicate deliberate acceptance of terms, with verification of an individual being left to other processes (e.g. IP address, MAC address, linkage to a specific email account, etc.), so forgery is less of an issue.
I have also seen "signatures" amount to checkboxes and "I agree" buttons. Generally, the higher the stakes and "more legal" the agreement, the more likely to these have been the "typed signatures" that you describe, but this seems to be decreasing in frequency, suggesting that its purpose was to stop gap a hole in legal acceptance by judges/courts/laws with regards to electronic communications.
Addendum: It should also note that the replacement of seals by signatures is not universal; for instance in Japan, seals are still used over signatures in the majority of cases.