Transparent Fractional-reserve banking as an alternative to the current system
As mentioned by @BrianRomanchuk which system is "better" is matter of opinion as that includes value judgement. For a radical ascetic a system which forces people to live on a subsistence level is the best system possible.  However, what you propose is most likely not optimal from economic perspective. 

What you propose here is basically transitioning to some sort 100%  fractional reserve system. There have been proposals for such system in the past - it is called 'the Chicago plan'. The idea first gained some traction in 30s at the University of Chicago (hence called the Chicago plan) and few years back the plan generated a bit of a buzz when Jaromir Benes and Michael Kumhof (IMF economists) published their paper Chicago Plan revisited where they tried to revive the idea. However, the idea was not really embraced by the profession. There are several reasons why.


You are right that if there would be 100% reserves and banks could lend only money that are in investment accounts (what you call setting aside), it would prevent the chance of bank runs (at least for the deposits itself). But evidence shows that deposit insurance by government does this equally well. You might not know that but in most developed countries including US all banks are by law required to have insurance for their deposits. Hence, even though your deposit money are technically being lend out you will not loose them if bank goes bust (at least up to certain limit, in the US the limit on deposit insurance is \$250.000, see here). Hence unless you are very high up in the income distribution you never have to make bank run as your money is safe, only if you are quite rich you might worry about it, but if you are rich enough to have such balance on your deposits there are other ways that you can protect yourself from this, and generally you would probably have some wealth manager who actually invests money for you. The only advantage of Chicago plan would be that there are always some 'shadow banks' that try to get around the deposit insurance rules by disguising themselves as other types of financial institutions, but it is generally believed this problem can be controlled by appropriate regulation. 
Fractional reserve system has many advantages relative to the full reserve system. In a fractional reserve system money supply can expand and contract more flexibly based on what is the demand for loans in the economy and consequently government through the central bank can easily manage money supply just by setting interest rate instead of actually changing monetary base. However, in your system that would no longer be possible and central banks would have to manage money supply through changes in monetary base which is much harder to do with the same precision as when you just play with interest rates. Managing quantity of money in economics is extremely important for mitigating the boom-bust cycles. Hence, even if we would agree for the sake of the argument the Chicago plan could help mitigate some financial crises it could make non-financial crises more difficult to cope it.
Under the 100% reserve plan there would be, in  the most likelihood, an overabundance of sterile deposits. Money that are just deposited but never lent out to help finance investment which is extremely beneficial for firm creation, job creation and economic activity in general. In such system firms would have tougher time to invest into new factories R&amp;D and so forth as most national savings would not be put into productive use. 


Hence, generally mainstream economists are not very cordial to the 100% reserve system and it is believed such system would be worse for the economy as it has some serious drawbacks and only some tentative possible advantages. 

The only country brave enough to try to implement such plan was Iceland, but to the best of my knowledge even though they announced in 2015 they want to implement it they did not really do it yet. According to most recent source I managed to find on their reform from 2016 it is still being reviewed by the committee set up to examine such system. 

However, let me reiterate that again what system is a best is a matter of opinion. A system that delivers the highest possible level of material welfare might not be the best system for everyone. You might oppose fractional banking on moral grounds etc. but most of the profession supports the fractional reserve system, or at least does not really endorse any alternative as better. In fact as an undergraduate economist the only other monetary system you will hear about is the gold/silver standard and even that probably in history class - by this I am just trying to show that among mainstream economists fractional reserve is really taken so for granted that alternatives are often, for better or worse, considered not worth to even mention to undergraduates.