Can you amend a contract?
Depending on the complexity of the contract and the part(s) you wish you change/add/delete/correct, you may wish to amend a contract or rewrite a new contract. Should you choose to amend, you have options.
The easiest way to amend a contract that has been printed and is awaiting signature is to pencil in a change and have both parties to it initial the written-in change. This occurs all the time and can be for substantive items or to correct typographical errors. That said, technically, this is not an "amendment" to the contract.
As noted in a comment on your question, to unilaterally alter the document and send it back for approval would amount to a rejection of the an offer and a counter-offer.
This solution applies mostly to contracts between private parties, but not necessarily only to them. In business contracts, the terms will often contain the method by which the contract may be amended. An example that appears often is a requirement to make amendments signed and in writing.
Lists of best practices on various websites include typing the amendment out on a separate document, labeling it distinctly as an amendment to the existing contract, indicate that the change contained in the new document is the only change being made to the original contract, and, of course, both parties must sign and date the amendment.
Would it be a good idea or a bad idea to make an amendment to the contract?
Whether it's the contract in the question you linked to or any other contract, the answer to this question will be specific to the factual issues related to the contract and the change being contemplated and the answer may not be the same for both parties. So, there really isn't a general answer to this question.