Realistically, you're going to need to show them everything. The bank isn't going to give you $300k without a pretty complete accounting of your financial position. There isn't going to be a lot of personal financial information that you're going to be able to keep away from the bank.
When you fill out the paperwork to apply for the loan, one of the forms you'll fill out requires you to confirm that you've supplied information about all your income, assets, and liabilities. Intentionally holding back information is only going to raise suspicions. Almost certainly, the bank will require tax transcripts from the IRS that show your income over the past few years to ensure that you have the income to pay back the loan. That's going to give them a pretty good map of where most of your assets are. They're going to require a credit report that will show most of your liabilities. If they see that you've reported thousands of dollars in dividends and capital gains from an account at Bob's Discount Brokerage and you don't provide a statement from that account, the underwriter will likely ask about it. If there are open accounts on your credit report that you haven't provided statements for, the underwriter will also ask about that. And once the underwriter has to start asking about missing accounts, they're likely to get more picky about your application than they would have if you had provided everything up front.
Most of the time, when people fail to list a non-trivial financial account, income source, or liability it's because they're trying to hide something. For example, an applicant might want to hide the fact that they have a large margin loan at Bob's Discount Brokerage or that they've got a large source of income that doesn't show up in their tax returns that is related to some illegal activity. Banks can be liable if they don't do their due diligence in detecting money laundering or if they misrepresent the borrower's debt-to-income ratio to the (usually government) entity that buys the note and/or the mortgage so they're going to need to see your full financial position.
As for why banks are going to ask for IRA statements when you have "enough" assets already documented, it comes down to product qualification and efficiency.
Mortgage banks in general tend to have a lot of different mortgage products available. It's not a simple "thumbs up"/ "thumbs down" decision-- different mixes of assets, income, and liabilities may end up getting approved for different programs with different rates or fee structures (for example, there are a number of different appraisal waiver programs that can reduce or eliminate the appraisal fee). When the computer runs through the options and tells your mortgage broker or loan officer what you qualify for, it generally has no way of saying that you could have qualified for a slightly better program if only one of your ratios was better. So your broker/ loan officer is going to want information on all your assets to make sure they can offer you the best deal (particularly if you are shopping around).
One of the efficiency metrics that every mortgage company is going to track is the number of times that a file needs to get touched. It takes an underwriter (or a loan officer, a closer, or anyone else) a decent amount of time to pick up a loan, figure out what needs to be done, and do it. If the underwriter doesn't get all the information they need up front-- say, they see a recurring $400 transfer from your checking account every month that you say is going to an IRA but they don't see a corresponding IRA statement proving that those are IRA contributions and not payments on an undisclosed liability-- then the underwriter has to go back and ask for them. When you upload those documents a day or two later, a different underwriter may pick up the file and will have to figure out where the first one left things in order to clear the condition. That takes time which costs the bank money (and means that everyone pays slightly higher rates) so the broker/ loan officer will want to get everything up front. It's far better from their standpoint to have statements that no one ends up needing than to have the underwriter come back later asking for additional information.
On the positive side, though, since mortgage banks have to collect such extensive financial information, regulators and banks take privacy very seriously. Nothing can completely eliminate the risk that your personal information will be misused. But mortgage banks are among the safest custodians of your personal information that you're going to find. A large mortgage bank is getting audited constantly by state and federal officials and is almost certainly retaining third party auditors to ensure that its policies meet all the data privacy laws and to attack their systems to ensure they don't have any vulnerabilities.