Your player has the right of it, mostly
All of the following information can be found for free in the D&D 5e Source Reference Document PDF as well as on the official 5e online supplement D&D Beyond. Page references are included for the information in the SRD PDF. All of the content quoted from the SRD can also be found in the core rulebooks.
Paladins and clerics both say the following under their spellcasting class feature (SRD pgs. 16, 31)
You can use a holy Symbol (see "Equipment") as a spellcasting focus for your [cleric/paladin] Spells.
The general rules for spellcasting foci as related to material components are as follows (SRD pg. 102):
A character can use a component pouch or a spellcasting focus (found in “Equipment”) in place of the components specified for a spell. But if a cost is indicated for a component, a character must have that specific component before he or she can cast the spell.
In particular, the holy symbol states (SRD pg. 67):
A holy symbol is a representation of a god or pantheon. It might be an amulet depicting a symbol representing a deity, the same symbol carefully engraved or inlaid as an emblem on a shield, or a tiny box holding a fragment of a sacred relic... A cleric or paladin can use a holy symbol as a spellcasting focus. To use the symbol in this way, the caster must hold it in hand, wear it visibly, or bear it on a shield.
These are the only rules as far as the book is concerned. Clerics and paladins can use a holy symbol as a spellcasting focus to replace the non-costly, non-consumed material components of their spells. No stipulation RAW is made that the holy symbol must be of an exactly certain quality, type, or even a match to the deity that the cleric or paladin worships. From these quotes, we can determine that:
For spellcasting, the holy symbol is only required when you need material components
The holy symbol is a spellcasting focus. Spellcasting foci by design are only used to replace non-costly, non-consumed material components. If your player is casting a spell without material components, they do not need their holy symbol at all and can cast it normally. If the player can provide the actual material component, they can also cast it normally, even without the spellcasting focus.
speaking of material components...
Component pouches can be used by anyone
The rules for component pouches are as follows (SRD pg. 67):
A component pouch is a small, watertight leather belt pouch that has compartments to hold all the material components and other special items you need to cast your spells, except for those components that have a specific cost (as indicated in a spell's description).
There are no other rules for pouches, aside from those general "spellcasting foci" rules I quoted above. Any spellcaster can use a component pouch for basically the same purpose as a focus. More on that topic can be found in this question.
If the player gets a component pouch, they can use it to cast spells that they would normally use their holy symbol for. The flavor is that the character stores the actual non-costly material components in the pouch. So, as long as the character has access to the pouch, they are actively providing those material components that a spell calls for. This, as was described above, allows the character to cast spells that require material components, even without a focus.
However, non-spell channel divinity features may require holy symbols
For example, the cleric's turn undead channel divinity says (SRD pg. 16, emphasis mine):
As an action, you present your holy symbol and speak a prayer censuring the undead.
If the player has any non-spell action such as turn undead, they need their holy symbol to use it. In this case, your cleric cannot turn undead until they have a replacement holy symbol, since they have no holy symbol to present.
Additionally, these options are not spells, so the component pouch cannot be used as a substitute in these instances.
How they replace the lost symbol is up to you.
Prices for holy symbols are found on the general equipment list (SRD pg. 69), so in theory it could be as easy to replace as going to a friendly church and buying a new one. There are no rules for "blessing" a holy symbol or the like. The description of the holy symbol item does say they can be "carefully engraved or inlaid as an emblem..." so it would imply that the symbol is more complicated than "crudely carving it out of a piece of wood", but ultimately, the DM is the one who gets to make the call on if such an attempt would work or not.