TM has been working fine since Leopard in 2007. (Do you really still use the Tiger MBP...?)
The Mac running Tiger (Server) is a B&W G3. It hasn't been on in a while, but it was running for several years. Next is my 17" PowerBook G4 running Leopard. I have two 2009 Minis running Catalina and a 2006 Mini running Snow Leopard. There is a 2006 17" MBP running Snow Leopard, a 2008 MBP running Sonoma and my 2009 MacPro running Sonoma.
From 2017 to 2020 my primary Mac was a PowerMac G5 Quad.
From the rest of what you say, you've already got a NAS setup -- so is your problem just not having the backups done automatically?
As said, there are any number of apps that will manage that for you.
Most of my difficulties came from around 2014 to about 2018 when the primary Macs in the house were all PowerPC and limited to Leopard max. There were eventually Intels in the house and that max went up to Snow Leopard by 2018.
• Time Machine had to sometimes be forced to use a network share for backup.
• Time Machine would initiate a backup to a network share, creating the disk image but erroring out. I was usually forced to first create the disk image, then have TM start the backup.
• If the network dropped out on the Mac backing up via TM, it would often force a full backup the next time around. This became a problem on a Fast Ethernet network with shared USB 1.0 drives.
• I got a lot of unable to complete TM backup errors. That usually resulted in having to recreate the backup and a full backup ensued - again.
• MOVING a TM backup is difficult. Finder says you can't move it because of permissions, etc. Sparseimages are easier to move, but TM makes sparsebundle images. Sparsebundles really do not like being moved, particularly if you're trying to move them while connected to the share they are on via SMB and not AFP (although both have problems). I can of course choose a new location to backup to with TM. Which at the time, on a Fast Ethernet network, meant another half day to full day time period for a full TM backup. If it completed and didn't error out somewhere in the middle.
By 2018 I have fully ditched TM and am using CCC to make backups. I also updated to a Gig-E network. In 2020, I am now primarily Intel but continue using CCC backups. At this point, most of them are going to an eSATA RAID drive shared by the B&W G3 (then my server). But by 2023 it's been retired and the 'server' is a 500mhz PowerMac G4 with two large internal SATA drives.
Late 2023 and I'm now using two NAS boxes, still with CCC.
It's now 2025 and because I've been using CCC for backups for quite a while, I can't speak to how TM functions on more modern Macs. The YOUNGEST Mac I have is a 2011 MBA. It's running High Sierra. But I've detailed my problems with TM in the past above.
So, if TM has gotten much better about things then I just have no current experience with it. All I have to go on is my past experience using the hardware I had at that time. It wasn't pleasant. Nor was it 'just plug it in' and 'set it up'.