As far as I understand, Taproot enhances privacy by allowing someone to only reveal a specific key path, which in my understanding means that it's fairly flexible - is it possible to construct a P2TR address that'd be spendable by someone with (knowledge about the possibility and) "access" to an existing P2PKH address?
Short answer: no, at least not via the key path (and using a script path would be less efficient than using P2WPKH instead)
Longer answer: no, and even if it were possible, this would be a terrible idea. The address a receiver gives you is how they expect you to pay them. Giving them money by paying in a different way is not something you should expect them to honor. Specifically, their wallet software may not support taproot, making the money unspendable for them. Even worse, their wallet could include a hardware security module or other security mechanism that can't just easily be upgraded to support taproot.
If they want you to send to a taproot address, they'll give you one.