I know it is partly my fault because I tend not to prioritise Influence buildings, but I find by mid-game when I actually want to negotiate tech or trade alliances the cost to maintain them each turn is prohibitive. Most pops are not auto-generating influence so spending one or two influence per turn per pop drains my influence pool really quickly. Am I missing something? Is it worth investing in lots of influence buildings and techs to be able to sustain trade or science treaties? I don't know how anyone could afford, say, a tech treaty with two or three other empires!

Is there an easy route to influence-building for races that aren't centred around it?

Some factions struggle with that, Ive found that Vodyani, UE, Horatio and Nakalim have no problem with influence. That said, getting more influence definitely helps for keeping up laws and agreements. If you plan on setting alliances and having science and trade agreements going, you should definitely invest into influence buildings early.

Now on the topic of importance. It's a small boost to what you already have, and it ramps up with more turns you spend on it. Personally, trade agreement is better than scientific, since other empire's subsidiary can give you luxuries you don't have as well as a trickle of science on its own. I didn't find science agreement all that impactful. Now cooperation protocol give a percentage boost which will probably be ths better option sometimes. It all comes down whether you want it or not. Choose which agreement you think you need, juggle laws around so you dont spend as much influence on them.

Besides the buildings and planet focus, I tend to manage it with Hero perks and careful selection of Laws.  Laws that cost influence per population can eat up your gains, and if you're not always benefiting from the effect, it may be better to abolish the law and vote it back after you've compensated.  Heroes that give bonuses to influence (they usually have a skill or two for increasing influence per unit, and then a flat percentage per system) can be stuck on a system with the influence buildings (ideally with a planet or two that generates influence from an anomoly or nearby space feature).  That being said, there are some factions that just won't be generating much to begin with... Cravers have almost no reason to research influence buildings, but might still want to be able to use a Diplomatic Imbroglio now & then to keep pressure up -- once their fleets have good captains, they may want to build up a governor hero that can help increase diplomatic options.