You could do it all with one on Paxton but as pointed out it wouldn't conform.
Should be an interface local to each door breaking the power directly like with local emergency door releases.
Can't rely on the controller or the access network in case it FTO
They were garbage decades ago, programming would go loopy at random.
It's not the electricians fault, no system should be affected from a power loss.
It wants decommissioning or replacing, even if you fixed the programming it's knackered and unreliable.
Ok so the long version...
If the power being removed triggered the alarm then the battery and most likely the charging circuit on the panel is faulty.
At this point the external sounder should sound, if it didn't that is also faulty.
You then also seem to have a tamper fault which could be related to poking about in the panel or the faulty battery or maybe something else.
All these things need a multimeter and some knowledge how to use one to confirm and pin point the issue and clear the lockout.
Systems that have been unmaintained for long periods of time are usually best to be re-evaluated and upgraded.
In short it sounds like there are a few things not working correctly. I'd either remove it or get someone to upgrade it with the building works in mind.
Ah a question...
Yes I'd say most old ADE stuff I see is tired and has either no or very low charging rate.
Every battery replacement should come with checking the charging rate and off mains change over, it's pointless and wasteful throwing a new battery in for it to drain out.
If the system was on engineer reset for tamper events then the installer code will need to be used to reset the system.
It's possible you still have a fault with the charging circuit or the standby battery, which needs testing with a multimeter.
DIY homeowners get all het up when they feel entitled to an engineers code...
* The programming of a system is owned by the programmer. The code protects the programmer from liability that programming was tampered.
* It's part of the regs that an end user shouldn't have access to the programming parts of there own system.
You are more than welcome to serve your time as an engineer to learn how to fully program a system.
The fact you think an alarm systems important feature is push notifications is concerning.
If it was sold with a working alarm system then take it up with estate agents. Something that should be checked when moving IMO.
No-one is going to come out and make it audible only for free and regulations require it to be programmed like that for signalling.
A working system will always need maintenance. Some people find someone to throw whatever junk in, annoy the neighbours when it doesn't work and bin it when it breaks.
Many just don't realise it needs maintenance until its unreliable and the electronics are now burnt out.
Shop around, I would say what you have been quoted for maintenance is over double what I'd expect.
I would argue it's as likely if not more likely to happen. Internet stuff has 24/7 uptime and can be accessed from anywhere in the world with minimal interaction.
A kid throwing brick would need to choose your house out of all the other houses in the village/town/city.
Point being security is all about layers against risk, if you have a backdoor in the layers what's the point in paying for the outer layers...
Someone maybe able/willing to takeover your system and make it audible only clearing the errors and allowing user reset but if you need to replace it just do that.
If you want the system replaced professionally then you don't need to do anything with the existing system. They are not designed to be easily disabled.
If the system design is fine then wiring and detection could probably be re-used.
The installers code is not for the user mainly due to liability no installer with any sense would do this, for this reason it also doesn't conform to standards.
Issue with integration is what the system can control, if your HA was compromised would they be able to disable the alarm.
Most have there own app now if that's your thing.
Shouldn't have been fitted like that, maybe if it had it's own way in the consumer but not on the lights.
The state of that consumer unit who knows what fun the electrics are.
If the alarm system is completely powered down when you have no power then both the usually external siren (SAB) and internal standby battery are faulty and need replacing.
Replacing the external siren usually requires access, replacing the internal battery requires a multimeter to check the system is not faulty and charging your new battery.
Anything less than this your not necessary going to have a working system.
If you don't want it disturbing the neighbours do them a favour and just power it down until it's fixed properly.