Time Capsule 5th Gen 3TB - how can I connect to existing wireless network?

Apple designed their routers to only work with another apple router. The exception is the Express but it works in a totally different way and has a different chip so you cannot duplicate that method.. I have tried.

The Apple router MUST connect by its ethernet port to your existing network.

If the TC must be located near the MBP.

And you cannot run ethernet then you can use powerline adapters as alternative that work for some of the people some of the time.

Or you can use a wireless to ethernet bridge or client device like the Express.. but there are other far superior bridges available.

Or buy a Thunderbolt/USB to Ethernet adapter.. which than then be used exclusively to the TC by ethernet while you still do wireless to main router for internet connection.

If the TC does not need to be located near the MBP

Plug it directly with a patch cable to your main wireless router .. wherever that is located. Turn off its wireless assuming your main router is AC wireless with good signal where you locate the MBP. Then you can access the TC via its network.. there are a few more tricks to make this reliable.. but you should be able to easily setup this way.

Using two wireless hops.. e.g... wireless to a bridge and then wireless from TC to laptop is to be avoided if at all possible. It will dramatically increase latency on the network. One wireless hop is more than enough. Powerline adapters work quite well but if you go that way.. buy a set from a store that allows returns.. I suggest AV1000 or faster types.. they are readily available in AV2000 spec now.. but those speeds are more fictional than fact.

is it even possible to connect 5th Gen Time Capsule with existing wifi network (so it can work without any ethernet connection) and there are no other apple devices in the wireless network.

No

When I try to use the option "Connect to existing wireless network" during TC setup after physical reset I'm redirected to a screen to create a TC wireless network (which I don't need - I just want TC to be a client in my existing network).

Connect the Time Capsule to your existing router using a permanent wired Ethernet cable connection and set up the Time Capsule to create a wireless network. If you don't need another wireless network for "guests" to use, you can simply turn off the WiFi function on the Time Capsule and it will operate as wired hard drive.