If you want to make a one-off , a single item, just for your own needs, then by all means, use a 5v 10A power supply, or whatever power supply you find, because it doesn't matter.
But, if you're thinking on making several "kits" like this, you may find that power adapters with higher voltage and lower current are cheaper than adapters with low voltage and high current - a 12v-24v 2-3A power adapters will be much cheaper than a 5v 10A power adapter.
The higher the current, the higher the losses in the cable between the power supply and the actual connector on your product.
For example, here's what I could buy from TME.eu today (you can change the language in top corner):
wallwarts :
6.03$ : 24W 12v x 2.0A
https://www.tme.eu/ro/details/posc12200a-h/alimentatoare-cu-stecar-integrat/pos/6.03$ : 24W 24v x 1.0A
https://www.tme.eu/ro/details/posc24100a-h/alimentatoare-cu-stecar-integrat/pos/6.55$ : 24W 15v x 1.6A
https://www.tme.eu/ro/details/posc15160a/alimentatoare-cu-stecar-integrat/pos/8.50$ : 36W 12v x 3.0A
https://www.tme.eu/ro/details/posc12300a-h/alimentatoare-cu-stecar-integrat/pos/8.50$ : 36W 24v x 1.5A
https://www.tme.eu/ro/details/posc24150a-h/alimentatoare-cu-stecar-integrat/pos/Now let's look for 5v wallwarts capable of same watts :
26.8$ : 30w 5v x 6A
https://www.tme.eu/ro/details/sga60e05-p1j/alimentatoare-cu-stecar-integrat/mean-well/Same deal for laptop adapter style supplies - you won't find 5v laptop adapter style supplies:
7.13$ 24w 12v x 2.00A
https://www.tme.eu/ro/details/posc12200d-c8-wh/alimentatoare-tip-desktop/pos/ 9.05$ 36w 12v x 3.00A
https://www.tme.eu/ro/details/posc12300d-c8-wh/alimentatoare-tip-desktop/pos/12.34$ 60w 24v x 2.50A
https://www.tme.eu/ro/details/posc24250d-c14/alimentatoare-tip-desktop/pos/11.16$ 45W 20v x 2.25A
https://www.tme.eu/ro/details/ak-nd-50/alimentatoare-pentru-laptopuri/akyga/cpsunotaky-07712/12.52$ 65W 19v x 3.42A
https://www.tme.eu/ro/details/ak-nd-01/alimentatoare-pentru-laptopuri/akyga/cpsunotaky-07051/A switching converter will convert the higher voltage to lower voltage with some efficiency, usually it's over 85% ... let's go with 90%. That means that 90% of the wattage will be available after conversion.
So for example, if you go with 36w 12v power adapter and you convert to 4.5v ... 90% of 36w is 32.4 watts and if you configure the regulator to output 4.5v, that means you'll have up to 32.4w / 4.5v = ~ 7.2A of current.
Instead of spending maybe 20-30$ on a 5v 10A adapter, you spend 8.5$ on higher voltage adapter, and maybe 3-4$ on the switching regulator and the other components it needs (inductor, mosfet maybe) and you also gain flexibility - you optimize the switching regulator to work with 12v but if you run out of 12v adapters, you could pair it with 15v or 24v adapter and it will still work, and your product will also work if the user accidentally uses the 7.5v or 9v adapter from another device with your product.
You say you have those TP4056 modules already. That's a linear regulator charger IC ... it only needs an input voltage a bit higher than 4.2v (the maximum lithium battery voltage) in order to properly charge it. Whatever extra voltage you give it will be dissipated as heat on the chip. So, for example, if you charge an almost discharged battery (let's say it measures 3.8v) at 1A of current, and you power the module with 5v, you'll have at least (5v - 3.8v ) x 1A = 1.2 watts as heat produced by the charger chip. It's just wasted energy.
If you have the ability to set the voltage a bit lower - 4.5v is a good value as it's around 0.3v higher than maximum needed - you'll get less losses in the form of heat and more modules would be able to charge at higher currents at same time.
Each module will pull current as needed by the battery - it depends on how discharged the battery will be. It's not a constant 1A of current. When the battery is all discharged, there's a period where the charger will give it some low amount of current until the measured voltage goes above some threshold, then the charger IC will push up to how much you configure it, and as the battery gets close to being fully charged, the current amount will lower.
You can see that graph in the bottom right corner of page 1 in the TP4056 datasheet :
https://dlnmh9ip6v2uc.cloudfront.net/datasheets/Prototyping/TP4056.pdfSlow charge until the voltage on battery goes above around 3v , then straight to 800mA and higher, then as battery gets to around 4.1v the current starts to decrease until the battery reaches 4.2v
If you plug 16 almost discharged batteries at the same time, you'll have 16 modules each trying to pull 1A of current, but you don't have 16 x 1A = 16A available. if the modules pull too much current, the power supply may go into protection mode, over current protection or over power protection, and may shut down.
For your own use, you could just let it be and just put a label on the box saying don't charge more than 10 fully empty batteries or something like that.
For a proper device, you have several options.
You could use a higher wattage supply (ex. a 60w adapter) and a strong switching regulator or several smaller switching regulators (ex 2 3$ ones, 1 for each group of 8 batteries or 4 $1 ones, 1 for each group of 4 batteries) to have the maximum current available for all modules.
You could have some sort of microcontroller that monitors how much power each module consumes and adds all up and if it's too much power it could disconnect some modules for a few seconds - for example power modules 1-4 and 9-11 for 10 seconds, then turn off and power modules 5-8 and 12-16 for 30 seconds or power 3 out of 4 groups for 5 minutes at a time - you get less power consumption but longer charging times.
You could have your microcontroller detect how many modules are active by measuring the power consumption each one has, and if the total power is too much, microcontroller could change the programmed maximum charge current set on each module (by connecting or disconnecting a resistor using a mechanical relay or other methods)