Is there an alternative metric to GDP that measures a country's assets rather than flows?
You are correct in pointing out the flaws of GDP. But honestly, it's the best we have came up with; measuring the stock of wealth is just impossible. 

You say you can estimate the value of your own assets, and of course it seems that way. We all feel that way, until we actually try. How much is your cellphone worth? Easy to know, go to ebay and look for similar phones in similar conditions and you get a price for that phone, right? Why don't we do that for all the country's cellphones? Well, because if we put all the cellphones in the market prices would drop down dramatically. Now, we're not obviously sending all the cellphones to the market, we just want to know how much its value is, but the point is that in order to have an estimation, we have to make assumptions regarding the market. The value of one cellphone is easy to calculate because it doesn't really affect the market. 

Additionally, who says that you have to sell it in your local market? Why don't you offer it in Australia or Japan or Algeria or wherever the price is higher? And why not sell it another day when the price is up? The same goes for your car, house, computer, clothes, books, etc.

Think now of your financial assets. You can know that, right? All one needs to do is see what the bank says, and it says that my investment portfolio is worth \$100.... at least today... and exactly at the time I checked. But remember: the value of any financial portfolio is going up and down with high volatility even in the same day. Now try to figure out the value of the financial assets of all the country. Well, you can do that. But again: just today, and just at the time you checked. 

Even though it doesn't seem like it, the same goes for goods. Take oil. Say that there is an oil reserve with x millions of barrels just waiting to be extracted and sent out to the market. How much is the value of that reserve? It depends on when you ask, whom you ask to, and who else is asking the same question (other sellers). What about gold? silver? copper? coal? wood? You can estimate a value in a given time, but that can be very different in another time.

What about land? Well, if I sow corn it will give me a given amount, but if I sow rice it will yield a different amount. Besides, would it be the same if my neighbor sows corn and I sow rice than if we both sow the same (competition)? What if I want to use it to generate electricity instead of growing food? Or what if I use it for livestock? What if I sell it? What if we all decide to do the same thing at the same time? Or what if we decide to do completely different things? Indeed to value farm land, you have to assume what use will it have.

Infrastructure is even more complicated. How much is a railroad valued? And what does value even mean in this context? There is no secondary market for railroads is there? There are opportunity costs, though. And we have the construction value too. But the we would have to make assumptions on its depreciation, and say how much of the revenue of the country is due to this specific segment of the railroad. To estimate this, one has to say how much the country would loss was the railroad not there (which is a fictitious value). What about parks?

Don't even get me started on natural resources (rivers, lakes, coast, jungles, biodiversity, etc). But even then, those are the easy ones. We still have left education, experience, skills and the like. But this post is long already, the point has been made and now I have a headache for thinking all this. 

Indeed, there are many serious and very clever attempts to make estimations of these. But they are used to answer very specific questions. GDP, on the other hand is easy: just sum all the sells of goods and services. We (economists) don't like it very much either, but that's all we have. Just try to accept it, and don't take it to seriously.