Just to lay out the problem, you have your Vtech modem/router (from your ISP) and a Netgear extender (from you). Your phone, when connected to your Vtech WiFi, will hang on to that SSID even when you've far enough outside its effective coverage and are within range of your Netgear's SSID (and presumably vice-versa). This sounds like a normal situation, as your phone is simply attempting to hold on to its working online connection until you manually switch it over to the other WiFi SSID.
Also, with both your modem/router and your extender having different SSIDs, that's what is creating the issue where you're not able to just walk around your home and stay connected without having to manually switch over the one with the strongest signal. Instead, you want your home network to have just one SSID for the WiFi. You mentioned using the same password for both SSID networks but that's not the issue. Your phone is detecting and connecting to the SSID, the network identifier, not the password, in this situation the password is just an incidental.
Leaving your modem/router settings as is, focus on re-configuring your Netgear extender so it's a) using the same SSID and same password as what your Vtech modem/router is using, and b) set your extender to be using a different channel than your modem/router. You referred to relying on a 'install assistant', generally those are focused on convenience and ease of use. You need to bypass those default settings and customize your WiFi to suit your particular situation. This leads into a particular caveat, WPS. Wireless manufacturers include this function because of its convenience factor, but they all disregard it has serious security problems. Consumers like it because it makes set up really simple, but at the same time it's compromising the security of a home network. Not a big issue if you live in a rural setting, where your closest neighbor might be down the road, but if you live in a more densely populated area, a condo in a big city for instance, than it's much easier for someone to exploit the WPS function of your WiFi devices. So if you see a WPS option in your modem/router or extender, only rely on it very judiciously.
https://www.howtogeek.com/176124/wi...-is-insecure-heres-why-you-should-disable-it/
https://en.wikipedia.org/wiki/Wi-Fi_Protected_Setup#Vulnerabilities
https://www.tomsguide.com/us/home-router-security,news-19245.html
But getting back to re-doing your Netgear extender, try going through the set up process again, from scratch, Do a Reset on your extender (see pg. 14 in the link below) and set it up again, this time using the
same SSID and password as your Vtech modem/router:
https://www.downloads.netgear.com/files/GDC/EX2700/EX2700_FullIG_12May14.pdf
Looking at the screen caps of that online PDF manual I can't tell if there's an 'advanced' tab in the setup page to set a specific channel but you should be able to in your Vtech's setup page:
http://099.net.il/wp-content/uploads/2015/12/VTECH-NB403.pdf
You might want to install this 'Wifi Analyzer' app on your phone. It's a really nice utility that will give you a graphical picture of what WiFi networks your phone is detecting, along with basic tools to measure aspects of your own WiFi network:
https://play.google.com/store/apps/details?id=com.farproc.wifi.analyzer&hl=en
Also, you might find this 'RouterCheck' app to be useful and informative. It does some fundamental tests on your current home network as far as security issues, but I think its best feature is its integral documentation, very well written and explains basic terminology for those who require it (it's not the most obvious though, there's a tiny 'find out more' link at the bottom of each topic):
https://play.google.com/store/apps/details?id=com.Sericon.RouterCheck.client.android
Finally, just so you get a better understanding of the channel issue, this link includes a nice, one-page explanation on WiFi basics (somewhat dated but still relevant).
https://www.geekzone.co.nz/sbiddle/8728