Why a bank would propose a depositor to invest part of their money to mutual funds, instead of bank doing it itself?

Banks investing their depositors money is extremely, and systemically, risky. What if those investments lose money, even if just for a little while? The bank could collapse and the depositors could lose their money, or could trigger a bailout from a government.

This is why - historically in the US - there is a legal requirement that banks can be only deposit banks that don't invest, or investment banks that don't take deposits, but not both. When a consumer invests with an investment bank, they understand that they might not be able to get their money back when they want, if at all. When people deposit money in a deposit bank, they expect to always be able to get all their money.

(I've simplified things and glossed over a lot of details here)

Banks are already 90-100% 'invested' in assets, mainly loans and also bonds. They have plenty of ways to attract deposits and to make investments.

Often, the bank's referral fee for directing customers to mutual funds is higher than the banks' expected profit for attracting the equivalent amount in deposits. Or, the bank is simply giving good advice to customers in order to keep them.

There are a few things going on here.

First, banks can't just invest all their depositors' money in mutual funds. Banking relies on the idea that when you deposit money today, the bank will definitely give it back to you when you need it. It's easy to take that for granted, to the extent that most of us choose banks based on their fees and customer service rather than an assessment of their financial soundness, and bank failures are comparatively rare in much of the world today, but they're not non-existent.

To prevent bank runs and bank failures and mitigate the harm they cause, governments have established bank regulators and deposit insurance schemes. These work hand-in-hand: capital requirements make sure that banks keep enough capital on hand to satisfy the needs of depositors and carefully regulate the amount of risk the bank takes on; deposit guarantees ensure that bank deposits up to a certain value, generally €100,000 in the EU, are guaranteed by the government if the bank becomes insolvent (this guarantee is only as good as the government's ability to back it up, but a failure of a government deposit guarantee would surely be a national if not global financial crisis, and at least you'd have lots of company).

But that means the banking sector is highly regulated. A bank can't, say, take their client deposits to the casino and bet it all on black at the roulette wheel, nor can they invest it in, say, a small biotech startup that may or may not be on the verge of developing a blockbuster new drug. Both involve considerable risk of loss, and it's the job of banking regulators to ensure deposits are kept safe and at low risk of loss. Accordingly, banks are required to comply with a complex set of regulations that define how much capital they need to hold and what kinds of investments they can buy to ensure appropriate levels of risk and liquidity.

All of that is a long way of saying that a bank can't just go out and buy all the mutual funds they want with their depositors' money. Bank deposits are regulated and guaranteed and everyone very strongly wants them to never lose your money, while mutual funds can contain all sorts of assets, are not guaranteed, and can and do lose value. If bank deposits were all completely backed by a pile of mutual funds, bank deposits would have the same risks as investing in mutual funds, and we emphatically do not want that.

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So that's why your bank can't just go buy the mutual funds themselves. But why are they so keen on encouraging you to do it? Negative interest rates. Banks are sitting on a lot of deposits right now that they may not particularly want. Bank regulators require them to put their capital in a lot of low-risk places that currently pay little to no interest or charge negative interest rates (i.e. the bank essentially has to pay to store money there). It's wildly unpopular with customers and politicians when banks start charging working people interest to keep their money safe, but some banks have reached the point they'd really prefer if some of their customers took their money elsewhere. It may be costing them money, or at least not making them very much, to keep your deposit.

If your bank is in that situation, they could run a "hi! it would be swell for us if some of you all would go find another bank" promotion, but people would find that weird and disconcerting at best and they'd prefer to shrink their deposits rather than their entire business. They could start raising fees and charging negative interest rates to their customers, but until quite recently, banks mostly haven't wanted to do that. Which means their best bet is to encourage their depositors to invest their funds in equities instead.

So the bank wins in a lot of ways. Instead of having to safeguard your deposit, which may be costing them money or not making them very much of it, they get to free themselves from that obligation and instead collect various commissions and referral fees from the sale of investment products (the mutual funds they're hawking may be funds that make the bank a lot of money rather than the best choices for your individual investing needs).