Simple way is to exchange cash for bitcoin with someone using p2p trading, this won't leave any trace.
For this you need to have someone in your local area who is interested in trading, and you can get connected in different ways.
Other way is by using decentralized exchanges and services like PeachBitcoin app, Bisq, Robosats, etc.

However what the first person said in his/her comment on this thread is the only legit way to me if you really want to get Bitcoin anonymously. Actually if you are interested to buy bitcoin and you do not have anyone in your local area that will still fit you, you can as well create a thread on this board that you want to get Bitcoin from someone through p2p. But, you will send money first before anyone will send you their Bitcoin. No body will ever send you Bitcoin because you are just coming to the forum.

Quite a lot of people are sceptical about face to face trades, because of the risks that come with it, but that is the only way you can trade your money for bitcoin without linking it with your bank account or identity. If you must use face to face trade, remember to meet your counterparty in a public place and you can also go along with your friend; also ensure the trade doesn't happen anywhere near your home so you're not followed.
That is exactly what OP wants to avoid.

There are people who provide these services and have a long history and reputation. They are not going to risk their reputation over your small amount of money. That said a lot of people end up doing face to face trades with complete randoms and such deals have a high chance of going sour the more money is involved. It is best to look for established P2P traders and not engage with strangers if you can avoid it.

OP, currently, there is no way you can stay 100% anonymous when buying crypto with fiat, if you choose to go one-on-one physical meeting, your face would be revealed to the seller, and such is very risky because you might be traced or trailed to your residence, except you traveled a far distance and agreed to use cash without any bank transfer at a neutral ground.

If you choose to use a bank transfer, your name would reflect, and through your name, you could be traced, and there is no way one could make an anonymous fiat bank transaction, which is never possible. Either way, it is a 50/50 scenario; you either risk revealing your face to your buyer if you want to go fiat cash payment, or you use a p2p, which you can transfer from your bank account with your name revealed.

Early years of bitcoin like 2014 or 2015 as you said in OP, this forum served well for that purpose and I also think that the forum is still good for this purpose. You can go to your local board or also in the global board and request P2P trading with any member of this forum, especially the ones that render the service. You can send first or vice versa, you don't have to worry yourself if you have seen the user's trust score and believe they are trust worthy.

If I read old threads, I saw how people traded and leave feedbacks, but today the feedback space has lost what it's supposed to be. Person can easily give positive feedback because they love the person without any evidence of trustworthiness.

Yes exactly we can use cash or gift cards or prepaid cards to buy Bitcoin without linking it to bank account. Even Bitcoin ATMs let us use temporary phone number for privacy. However even if our financial details are hidden but person seen on camera means someone could potentially figure out who we are as your funny idea of  $1 mustache shows. So while we can limit what is known about us online being physically present always adds another layer of things to think about for privacy.

Using cash and trading with strangers far away from where we live is probably the best way to ensure the highest level of anonymity for us. But be careful, this can be dangerous because we can be attacked. Meanwhile, using a bank account will reveal and store detailed information about our transactions.

In short, you are right, we can never be 100% anonymous. The only thing we can do is minimize information disclosure, increase anonymity with secure transactions. We cannot expect complete anonymity when we have to use fiat money, bank accounts and need a counterparty to buy/sell bitcoins.

P2P is a form of transaction or exchange of buying and selling directly between two individuals without an intermediary. That means P2P services provided by centralized platforms like Binance, Bybit...are not pure P2P by definition due to their intervention (intermediaries). In other words, exchanges are distorting the P2P concept and applying it to their centralized platforms, and that has been misleading many people.

P2P is the best way to stay anonymous and enhance privacy, but there are also certain risks such as scams, so we need to be careful when trading p2p.

Bitcoin is a decentralized asset and boasts excellent security. But when it comes to anonymity, I don't think we can be completely anonymous when buying Bitcoin. As you said, when we buy Bitcoin, the bank account we use to top up to an exchange or conduct P2P transactions will certainly reveal our real name or identity. So, with this in mind, truly remaining anonymous will be very difficult.

But I don't think that's a problem. The most important thing when buying Bitcoin is to find a truly reputable exchange, like Binance. With this, we can at least feel safer. Furthermore, if you want a little more anonymity, it's better not to buy Bitcoin using P2P methods, but rather to buy it directly from the Bitcoin market. Buying directly from an exchange is likely to be more anonymous. We don't know who we're buying Bitcoin from, and neither does the person selling it, who will likely have the same thoughts.

Then, speaking of people from 2014-2015 and earlier, as far as I know, they also bought Bitcoin on exchanges. Because exchanges already existed at that time. But perhaps before exchanges existed, I've read that people bought Bitcoin directly or through forums (like this forum).

So, the point is, even before, if we truly wanted 100% anonymity when buying Bitcoin, it was very difficult. But you don't need to worry, because even though it is difficult to be completely anonymous when buying bitcoin, I am sure our bitcoin will remain safe, as long as we can take good care of it.