We'll see if my reply helps at all. I believe one of the things you are asking is whether or not it is possible to open up a position in a gold-related ETF inside a 401k and what other types of accounts could work for that. The answer, based on my experience, is that MOST 401k plans due to design will not allow you to buy/sell ETF's and individual stocks. But I believe you were kind of leaning toward not going that route anyway. Discount and full-service brokers can set up various accounts for you (tax-sheltered, taxable, whatever) that will allow you to buy/sell gold ETF's all day long. For example, Fidelity Investments or Schwab can set up a Roth IRA account for you, you can fund it with $6k (max for 2021) and then have at it with buying/selling just about anything as they offer a variety of things on their platform. Any gains should be tax free. Any losses will not carry tax advantages either. Once you withdraw, there should be no tax but subject to some stipulations. Oh and not long ago, trading commissions were dropped by most discount brokers so you are trading almost fee-free (government fees still remain but they are pretty small).
While GLD is by far the most liquid gold-linked ETF in the States, I prefer AAAU and SGOL. I'm not 100% sure how much actual physical gold is behind any of them. No one really is. AAAU was recently sold from Perth Mint to Goldman Sachs, SGOL (and GLD?) use JP Morgan as custodian... We can have a long debate about that. If there is any it's somewhere far away never to be seen by small fish like us.
The biggest advantage of AAAU and SGOL over GLD is annual expense. They are "cheaper" if you hold them for over a year. But if you prefer to get in and out frequently then GLD is the best due to tiny bid/ask spreads and incredible daily trading volume. If you are really into it you can look at UGL which is a leveraged fund which "seeks daily investment results, before fees and expenses, that correspond to two times (2x) the daily performance of the Bloomberg Gold Subindex". GLL is the inverse of UGL.
Let us know whether or not this is at all headed down the right path...