Okay, this is really getting into the minutiae of how hotel operate. And while I haven't worked at a Hyatt, the following is how other chains operate. It's possible that Hyatt does things differently (although I kind of doubt it because it's mostly based on common sense).

There are two reasons why hotels always want their premium rooms to have a higher Average Daily Rate (ADR) when compared to standard rooms. The first reason is that hotel owners like seeing that those rooms are earning X dollars more than standard rooms on the daily reports. The higher the rate, the easier it is for owners to justify the cost to keep those rooms as well as the services (aka labor) necessary to provide an upgraded experience.

The second reason why it's desirable to have a higher rate in premium rooms is that any corporate-issued upgrade instruments (like chain-wide free-night certificates for a particularly dismal stay at another property) get reimbursed at the ADR of the upgraded room type.

As such, if there's a choice between a cash booking and a points one, the cash reservation is going to get a slight advantage. And I want to emphasize the "slight" part. Neither of a reasons listed above are consequential. Regardless of who gets upgraded, the total revenue for the night isn't going to change (nor are the labor costs, for that matter). And those corporate-issued upgrades are rare enough that they aren't much of a factor, to be honest.

Here's the huge caveat: A normal Front Desk representative doesn't know any of the above. They are only trained to follow the rules set in the Terms and Conditions of the chain's frequent guest program as well as any upgrade procedures at their hotel. That's why I described all of this as minutiae.

To me, this is the most important factor at play here:

Pretty much every decently-run hotel on the planet will prioritize repeat guests of their hotel over anyone else in terms of a tie-breaker for an upgrade.

It's probably worth noting that Front Desk staff does have a good deal of authority when it comes to room assignments and upgrades at most hotels. And even if an upgrade isn't available, all rooms within a certain category might not be equal. As long as they aren't new, the Front Desk clerk will probably know room X is near the kitchen exhaust unit, room Y overlooks the loading dock, but room Z has a 60% ocean-view even though it's categorized as a standard room.

So, treating the Front Desk with respect (as you should any service worker) is likely to be beneficial.

Hint for Frequent Guests of a Specific Hotel: If you get upgraded (or the Front Desk representative places you in a "nicer" room within the category booked), taking the time to return to the Front Desk to say a simple "Thank you, the room you gave me is great" can be beneficial. Front Desk representatives will remember you the next time you return. And an above-average representative will make a positive note in your account.

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TL;DR: Cash booking get the slight edge over points reservations. Being a repeat guest at the hotel is probably far more important.