The last 300 years have been a historical anomaly. In addition, that anomaly suffers from survivorship bias.
If you took some money 300 years ago, and invested it in the right kinds of assets in the right corner of the world, you'd easily be insanely wealthy today.
But pick the wrong corner of the world, and your investment would be worth nothing. Do the same thing 3300, 2300, or 1300 years ago, and your investment would probably be worth nothing.
The last 300 years has experienced multiple chained economic singularities we call steps in the industrial revolution. From water power and canals, to steam power and railroads, to oil and the internal combustion engine, to electricity and computers, each has been a wave of insanely fast change and growth.
In this environment, putting aside money to bet on the future being richer than the present tends to come up aces a lot. What more, doing so in the set of superpowers that won the global conflicts -- the East-Atlantic British Empire (UK), followed by the West-Atlantic British Empire (USA) -- is going to do even better.
You toss down a fortune in Germany in the inter-war years when it looked like it was going to become a superpower, and you'd have lost it all if you picked the wrong side of East/West Germany after the war.
The world is full of places where it looked like it was a good idea to invest, then within the century every investment would be lost.
What more, such investments would be extremely alien to someone used to the previous thousands of years of history. The idea you'd take wealth and hand it over to someone to make it bigger would be a short term thing as far as the immortals are concerned; every society crashes, and when it does your investments are worth nothing. What more, in most of history, holding onto your investments requires political and military power, as if you get rich someone else will just take your stuff.
And even after all this, your immortals wouldn't be much better off than random Dynasties. Dynasties who pass down assets to a single heir are economically very similar to immortals, especially if the previous heir gets to pick the next one. Such Dynasties where quite successful -- the top ones we know as "Kings" and other nobles -- but not universally so. As your immortals can suffer being killed, such immortal single-person dynasties would typically end with simply killing the immortal.
Long-lived immortals would be aware of this problem; those that have lived a long time will have adapted their lifestyles to avoid it.
The immortals might not have the "stupid heir" problem, but they are in a sense their own stupid heir. They'll get set in their ways and not adapt to the current situation, and their financial empire will fall.
In the post-industrial-revolution singularity states, some of these concerns don't apply. The need to guard your wealth with guns is a bit less (until your society collapses), the need to guard your wealth politically remains the same (or greater).
You'll still be about as well off as large dynasties that have passed their wealth down to one person. Your investment habits that are successful in one era will fail in another, and if you chase the rabbit and change with what habits have recently been successful you'll always miss the initial growth surges.
So your return on your savings will probably be, on average, no better than the financial giants of the past. In the UK, that would be the nobles; while they are (on average) not so badly off, they aren't the titans of the world economy you seem to imply the immortals are.
In the USA, similarly, the heirs of the robber barons of 1800s are not the top of the economic pile today.