Your iPhone contains technology that is designed to control the flow of current from a charger into your battery. The charger itself - as long as it meets some generic requirements for voltage regulation and current capacity - has little or nothing to do with how your battery performs over time and usage cycles. Apple refers to their technology as "Optimized Battery Charging".
This technology is implemented using a combination of hardware and software. The hardware may be sourced by one of the large electronics manufacturing companies; Texas Instruments and Analog Devices both offer a wide variety of integrated circuits designed specifically for charging batteries. These ICs meter out the voltages and currents needed to effect optimal charging of various battery chemistries. Some of these ICs also include serial bus interfaces for communication of telemetry and commands with the "outside world". Ultimately, however, it is Apple's software that makes all the "final decisions" wrt battery management.
Apple has chosen the Lithium-Ion battery chemistry for their devices. All battery chemistries offer trade-offs; finicky charging requirements are one of the significant ones for Lithium-Ion batteries. At the risk of over-simplifying, charging a Li-Ion battery is a two-stage process: there is a constant-current charge phase, and at some point that transitions to a constant-voltage charge phase. This article offers a more detailed description of the charging process. Still more details are offered in this Technical Handbook on Lithium Ion Rechargeable Batteries. Here's an overview of the charging process:
AFAIK, Apple does not publish exact figures for their batteries, but batteries in late-model iPhones are said to be ≈ 1.8 Amp-Hours. This is relevant to the OP's question because there are recommended limits to charging rates for Li-Ion batteries. Charging rates are generally quoted in terms of the battery capacity (C) in amp-hours; a charging rate of 1C would mean a charging rate of 1.8 Amps for the typical iPhone battery.
Most recommendations for Li-Ion batteries call for a maximum charge rate of 1C. It is also frequently stated that charging below this 1C rate improves the battery longevity, reduces battery temperature, etc. I was unable to find a specifc recommendation for iPhone batteries, but it seems quite likely they would also benefit from a charge rate below 1C. One source stated:
For safety reasons, most batteries should be charged at between 0.5C and 0.7C.
If this is applied to a 5-volt wall-charger, then the most current that should be drawn is:
Imax ≤ 1.8 A-h × 0.7C = 1.2A
And so anything more is likely to be unused. Whether or not it's possible to get 1.2A from a 5W wall charger will depend on its design and construction. If it's a simple transformer-rectifier, you will almost certainly be able to draw 1.2A. However, if the charger output voltage is regulated, you may or may not be able to draw that additional 200 mA. However, in either case, the only implication is that a 1.0 A limit may delay time required to reach a full charge by a small amount.
And so, the answer to the question "Will using higher wattage chargers degrade iPhone battery more quickly?" is this:
Under a safe charging strategy - not much. And we don't know exactly what Apple's charging strategy is, so only Apple can answer that question definitively. However, it can be said if Apple's charging strategy is aggressive (fast-charging), then it is possible that a higher wattage charger would enable Apple to deliver greater charging currents that may be deleterious to the battery's longevity.
Another thing to consider (Warning: Apple supplicants may be triggered):
Batteries are complex devices - chargers are not (relatively speaking at least). The veracity of this statement is underscored by Apple's mis-steps - past and present. One of the world's largest technology companies with virtually unlimited resources, and years of experience with Li-Ion battery technology has recently announced another significant battery issue:
This recent report in Forbes magazine will make anyone think twice about "upgrading" to iOS 14:
...the company states the only fix is to “erase all content and settings from your iPhone”
What's an Apple customer to do??? On one hand, Apple's advice for Maximizing Battery Life and Lifespan states the following:
Update to the latest software
But now we see that isn't necessarily the best advice!