It's fine, but choose carefully as they do vary: for example some tenements have plenty of space to leave a buggy downstairs if you want, others don't. (Beware of theft, though: given that bicycles get nicked fairly regularly, it has to be a possibility that an expensive buggy might, though it didn't happen to us. In practice I'd often leave it downstairs when I came in with DC and my strong DH would carry it upstairs when he came in later.)
It's great not having internal stairs and I think bed time is easier given that DC's bedroom is still in the middle of things: bedtime doesn't feel like banishment.
Many tenement flats are rented by groups of students, and you could get a noisy group so you might want to think about whether you have the right character to be assertive if you needed to be and not be emotionally destroyed by it. We've never had a really bad experience, though, and tbh the occasional noisy party helps me feel less worried about the nights DC cried or the music practice that goes on now!
One downside is that even big flats tend to have only one bathroom. Some have boxrooms that have been, or could be, converted if that's a big deal to you.
Most shared gardens are underused, but unless you are on the ground floor, going out to the garden with a small child is almost as much palaver as going to a nearby playground if there is one; we didn't actually use the garden much except when visiting children came once they were older. (That said, an advantage of being in the centre is definitely easy access to a choice of playgrounds.)
Main door's lovely, but you really notice how much extra light there is on higher floors, and higher floors also feel (and are) more secure: stuff like being able to leave a window slightly open when you're out in the summer is nice. Also being on a middle floor cuts your heating bills noticeably!
You have to bear in mind in budgeting that expensive things (roof, masonry, window frames, etc.) have to be done from time to time and then they have to be organised with others on the stair which can be a hassle: otoh, it's nice not to have sole financial responsibility for a roof and if someone else on the stair is a great organiser you may have to do no more than pay.
The tenement flats feel great to live in: good sized rooms, high ceilings, etc., even if you haven't that many rooms.