This answer kind of jumps around a bit, but I think it's all relevant info. There is a TL;DR at the bottom that more directly answers the question.
Distributed Ledger Technology is totally reliant on participation to establish the principle of Distributed Trust. If there are not enough independent actors in the network, you cannot rely on the immutability of the ledger.
There are actually two things that contribute to building distributed consensus over time, and the distinction between them is quite important in the context of your question.
The bitcoin network is comprised of bitcoin nodes, and in addition to that there are bitcoin miners alongside it. All miners will require a node, but not all nodes are miners. (though if we hold number of nodes constant, a larger portion of those nodes also being miners is indeed a positive in terms of attaining 'distributed trust').
So what is distributed trust then? There are a few parts to it. Immutability of past transactions is one part. Consistency of network rules is another. If the rules were expected to change, how might that affect your investment? You probably wouldn't want someone else to change the rules in a way which unfairly benefits them, but causes you detriment, right?
Miners are incentivized via the block reward to continue appending blocks to the chain, the result of which is a probabilistic immutability of past blocks, and an incentive to continue creating new ones.
Nodes are run by individuals who want to interact with the network without having to trust any third party (sovereignty is the incentive), and it is all nodes reaching consensus together which defines what the consensus rules truly are. If one node tries to change a rule, the other nodes will simply ignore them. This is how consistency in network rules over time is achieved: rule changes generally require very wide-spread consensus among network participants.
So if you don't have enough users incentivized to run both nodes and miners, then you are going to have a hard time actually establishing distributed trust. But perhaps you can still market your product with buzzwords and grab a piece of that apparent 'billions of dollars' ready to flow in..
...let's say some agency decided they wanted to seed a new public blockchain to do something like: verify that a particular coffee brand came from FairTrade sources. How would they incentivise independent actors to verify transactions in this chain?
Supply chain tracking is the perfect example of an often cited 'blockchain application' that makes no sense, but not because of incentives. It fails even before that: the end user looking up their product's history will need to trust that the company that made the product input truthful data to the chain. What is to stop the company from just making up data, that has no truthful relation to the product that the 'supply-chain-tracking-proof' has been placed on? And so if you must trust the company anyways... well then why are we jumping through all these hoops trying to carefully craft incentives to create a functioning blockchain?
This is known as the 'oracle problem', involving situations where data from the physical world is required to be input to the blockchain. You must trust the oracle which is in charge of inputting the information.
From where I'm sitting, its hard to see how a public blockchain can have any application beyond replacing fiat currency.
Timestamping data is one application that exists and is utilized, for example see opentimestamps.org This is done on the bitcoin chain without creating any token or whatever, which is perhaps why there isn't as much investor-hype on it (there is no timestamp-coin to sell, after all).
Namecoin is perhaps interesting in this regard too, because while it makes more sense on technical level, it apparently has little value in the free market. So this is another example where a blockchain solution doesn't seem viable, but for the reason of poor market-fit. People just don't seem to care about using it, at least for now.
TL;DR: Incentivizing network participation is hard, and the incentives need to align not just for miners, but for nodes as well. Removing the network's native asset drastically alters these incentives, and it isn't clear if there is an alternative construction that will restore the game theoretic balance that will ensure 'distributed trust' can be maintained. I would guess that such a system might not really look like bitcoin much at all.
Additionally, there are potentially many reasons a blockchain is not a good solution to a particular problem, that preclude us from even reasonably wanting to consider how we might construct a 'distributed trust' network to address that problem in the first place.