Let’s look at it from a different angle. One does not file a resignation before receiving a countersigned copy from the company.
To continue working at a company, you are usually expected to sign every piece of paper that they put in front of you throughout your tenure. If you don't, the consequences could range from absolutely nothing, to being pestered every six months, to being kept off a project or not promoted, to being fired.
If at some point you become popular or are seen as critical, you may be able to spend some of your political capital to get around signing some of the papers that you don't particularly want to sign, but this doesn't always work, and you can't always get away with this indefinitely.
An amendment to your offer of employment that tries to prevent you from moonlighting could come after your first day, or after your third month. They get to choose the timing, and the more unscrupulous will absolutely use this to their advantage. They won't give it to you after the first day or week, so you don't just walk -- they'll wait until you've settled in and are more likely to concede in frustration.
It seems like the better question is: are there other documents than this contract on which this job depends to sign and can I see them. If yes, can you please put this in the contract.
If you've signed a complete contract already, and there are new things to sign, they're asking you to agree to a new contract. Which you don't have to agree to, and if you don't, then anything you've already signed is still in effect. To get rid of you after that they have to resort to the termination provisions in the original contract. Of course "refusing to agree to corporate policy" is typically something that they can terminate you for. So a big part of the law dictating how much they can make you sign later is found in termination provisions. If your original contract's termination clauses are written so broadly that they could fire you easily for refusing to agree to a new contract that essentially rewrites the old one, then there are things you should see a lawyer about before touching it. (This is a much bigger deal than policy changes. I'm talking "no we actually have 10 more onerous pre-conditions to exercising stock options that we hid from you, such that you will find it almost impossible to do. sign here".) Large swaths of contract law are devoted to invalidating terms that say versions of "I can dictate the terms".
So my guess is that the horror stories were not supported by the original contract using likely-invalid language, but rather purely on the threat of messing up someone's new job. Especially if you're e.g. moving for a job, your goal is to sniff this out before you sign. So insist on full pre-signing transparency, get assurances that there are no other documents that are considered mandatory, and watch the responses carefully.
You could also being a contract of your own for them to sign. They won't, of course, but it can help drive home the point why you need time to look at the contract.
Story here not too long ago of a Google employee being let go on his first day because he did exactly what you're suggesting. I'm sure my company would do the same. When you hire over a thousand people a year (let alone thousands), they're not going to modify their efficient procedures for you. It's cheaper to hire another person.
I'm sure they'll let you take the contract home, but your only option really is to quit.
Corollary: continuing employment may depend on adherence to policies you don't control, regardless of prior agreement.
Not all law is contract law, and this remains true of employment law.
Like I was going to sign, give notice at my current job, and if they're not happy I simply don't have a job.
I nope'd out of that process. Their HR was a mess.
