##Your player has the right of it, mostly
Paladins and clerics both say the following under their spellcasting class feature
You can use a holy Symbol (see "Equipment") as a spellcasting focus for your [cleric/paladin] Spells.
The general rules for spellcasting foci are as follows:
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:
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.
###The holy symbol is only used for spells 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:
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. Any spellcaster can use a component pouch for basically the same purpose as a focus. More on that 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.
###However, non-spell channel divinity features frequently use holy symbols For example, the cleric's turn undead channel divinity says:
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, so in theory 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 seem 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.