Just not fit for running?

It sounds like you are doing more or less what is recommended for beginners to do in order to start running. That is essentially doing intervals - either by time or distance. Since you already have a walking routine you are comfortable with you can simply get warmed up by walking and then start programming in 1 minute jogging bursts into your 30 min walk. As you get comfortable with that (perhaps after a week, increase it to 1.5 mins or 2 minutes, then keep pushing moving up the increments weekly at a faster pace as you get used to running until you are running more than walking in a few weeks, and then until you are able to run for 30 min uninterrupted). The speed should be such that you can comfortably speak but not sing. That may very well put you close to a fast walking speed but perhaps with a somewhat different cadence/type of movement. As Lietchi said that is common, and you will get faster over time.
I have done both walking and running for health. I have been an avid hiker my whole life but always “wanted” to run. I tried many times and gave up because it seemed too hard even though I was in good shape and at a good weight with no physical impediments. For me it was a combination of the wrong approach (trying to run continuously and too fast at the outset rather than starting with a walk/run approach) and nothing particular to motivate me to keep going. I signed up for a Rock & Roll Marathon in a city I wanted to visit and started training with that goal in mind and it suddenly worked.
That said, I also gained weight training for it because I somehow thought running was harder or burned a lot more calories than walking - it only burns marginally more, and I was eating significantly more/differently and perceiving myself to be way more tired because it was a new type of exercise for me. I did not injure myself, but a lot of other new runners do, so make sure you get yourself fitted for proper running shoes (they are different than walking shoes) and do additional complimentary strength training exercises to keep your core strong so that you don’t injure your back or get tired as easily. I was also working out with a personal trainer to build more strength around my knees and thighs bc I have a history of knee injuries, strengthen my ankles bc runners often twist those, and improve balance and all over strength too through pilates and yoga.
I love running, but if I am honest I will always be a walker/hiker at heart - they are kindred pursuits and both are great, so you shouldn’t feel like one is somehow better or more desirable to pursue for health than the other. They each have their merits, both athletically and in other ways that promote well being.

I would whole heartedly recommend the Couch to 5k program, there are various versions of it but all pretty similar.
It gradually increases the length of the running intervals over the weeks, and of course you can repeat weeks if you need to.

I found the structure really helpful. Started by following it in my early 30s, from a very non-athletic background. About 10 years down the line am still at it. Have never cared about speed, go for distance goals instead.

It did feel very hard to start with, but after a few weeks I felt a lot less like I was going to keel over and started to enjoy myself. These days I like running trails, I go very slowly and plan routes that go home past nice cafes 😁. Good shoes help enormously, and socks make more difference than you might think.