As the sibling comment mentioned, this is a solved problem. MySQL and MariaDB take a very brief lock on the table at the very end of index creation that you will not notice, I promise. Postgres does the same if you use the CONCURRENTLY option for index builds.
If for some reason you do need to migrate data to a new table, triggers.
If somehow these still don’t solve your problem, ProxySQL or the equivalent (you are running some kind of connection pooler, right?) can rewrite queries on the fly to do whatever you want.
It really isn't. The addition of an index was an example, and a relatively straightforward/simple one, of general schema migration pain. And even index additions aren't a solved problem: CONCURRENTLY (postgres only) and other affordances don't work in every case, as their documentation abundantly illustrates--what about overlapping indices, foreign key "inherited"/implicit indexes, UNIQUE constraints, and so on? Even in postgres with a DDL expert on hand, things get hairy fast.
In MySQL/maria, that "very brief lock" at the end of index creation can turn out to be anything but, if the table is high-write-volume, since InnoDB locks aren't FIFO--at least, not in the way people expect them to be.
And again, that's a simple case for which mitigations exist! More complex schema migrations hit immense amounts of trouble at even medium sizes of data and query volumes.
Online-schema-change tools that use triggers/cutovers are extremely limited: not only do they take locks to set up and manage their triggers/replication systems internally (meaning that they can cause severe disruption, or fail to migrate, in the face of continuous $thousands/sec of writes to a table), but their underlying mechanism is also often one of duplicating a table entirely, bulk-copying data from old to new, using triggers to stream changes during the migration, and then cut over. That mechanism is flawed for large tables (2x the data might not fit), flawed for high write volumes (2x the update operations is an unacceptable latency cost in plenty of circumstances), and just plain risky (that final cutover/drop/rename to point things at the new table is and probably will always be extremely lock-ful).
To be clear,the availability of advanced database facilities like CREATE INDEX ... CONCURRENTLY, triggers, and online-schema-change tools are all good things! My point is that this is very, very far from a solved or painless problem space--one that starts to become quite painful at even moderate scale.
If for some reason you do need to migrate data to a new table, triggers.
If somehow these still don’t solve your problem, ProxySQL or the equivalent (you are running some kind of connection pooler, right?) can rewrite queries on the fly to do whatever you want.