The answer to this question is not entirely clear, as nobody can predict the future to see how successful the bitcoin network will ultimately be, and scaling to meet the financial demands of the world is quite the task.
That said, there is an important point to understand before exploring this question further: the number of transactions that happen on-chain do not map 1:1 with users making payments. The word 'transaction' is maybe a bit of a misnomer here unfortunately, it might be more appropriate to call it a 'settlement', as a single on-chain transaction can account for and settle a potentially unlimited number of payments. I think this article does a good job of describing this in detail.
What this means is that there are a lot of different ways that a user can store and transact value that is backed by the bitcoin network, without actually sending their own on-chain transaction! In some cases, the user may not even need to send an on-chain transaction at all.
Scaling via 'off-chain' solutions
There are a number of known methods to scale the number of transactions that users can engage in (see Murch's answer here):
- The lightning network (and other payment channel techs)
- Lightning channel factories
- Federated sidechains
- Custodial off-chain services
The general idea of these solutions is that a single bitcoin transaction can be crafted in a way that allows users to exchange value without sending further bitcoin transactions. Each of these methods requires additional trust/security considerations, but many users are willing to make these trade-offs for increased utility/functionality.
There are some other ways to transfer bitcoins without sending an on-chain transaction, a few of which are detailed in this answer. But some of those solutions (eg the OpenDime) are interesting novelties, more than widely employed solutions.
Scaling via 'on-chain' solutions
There are also ways to scale the number of bitcoin transactions that the network can handle, to copy paste from Murch's answer that is linked above:
- Blocksize/blockweight increase
- Faster blocks
- The witness discount of segregated witness
- Smaller size of Schnorr signatures
- Bellare-Neven signature aggregation
- Key aggregation
Additionally, batching payments can provide savings in both blockspace, and transaction fees.
Note that there are very important engineering considerations when attempting to scale the network by altering these on-chain parameters. As example, increasing the blocksize drastically can provide a linear increase in transaction throughput, but it also greatly increases the resources required to run a node on the network, which is damaging to the network's health. Without a large number of independent node operators, some of the network's most important properties (eg. censorship resistance) will be substantially degraded.
If that's the case, considered the recent bull crypto market, we can expect there is a high chance Bitcoin reaches the maximum number of transactions per day, which will cause the crash of the Bitcoin price and a big shift in the crypto world.
I don't think so, it seems that the current rhetoric of users (investors) is not "this is valuable because we can send a huge number of transactions!", but rather, "this is valuable because it is a scarce digital object with strong guarantees of censorship resistance and an incorruptible economic policy".
Of course, serving a larger audience of people is desirable, but just because the system hits its max throughput does not mean it will suddenly fail. In fact, such situations serve as great motivation for users to find more efficient ways to transact. As example, a recent period of network congestion lead to a great number of users turning their attention to the issue of poor fee estimation present in some wallets.
If nothing else, consider that even if bitcoin cannot scale to meet daily global transaction demands, it may still be an incredibly useful and valuable technology nonetheless! As analogy, consider that gold is extremely valuable, yet it is very rarely used in day-to-day transactions. (I don't mean to imply that this is the ideal future for bitcoin, I am hopeful it will continue to scale, but it is something to be considered in the extreme of OP's question "will it fail if it hits max transaction throughput?").