I don’t think there’s a single correct answer to your question (i.e., is a keyless tenor recorder a drawback). I suspect that it depends on the individual instrument and the player.
You might read through the “tenor recorders” guide on the Antique Sound Workshop’s site.
Some excerpts from that:
Some tenor recorders have a single or double key on the footjoint to facilitate closing the lowest fingered tone hole…Players needing a low c# should purchase either a keyless instrument or an instrument with a double key…
Tenor recorders vary a great deal more than smaller instruments in design and ease of playing…small bore tenors have, as one might surmise, a narrower bore toward the bottom, are shorter in length, and are typically keyless instruments…larger bore tenors are appreciably wider in bore toward the foot, are longer in length, and usually have a single or double key on the lowest tone hole…
Conventional wisdom in the recorder world would have one believe that players with smaller hands should consider only keyed tenor recorders; however, “it ain’t necessarily so.” Keyless instruments are smaller, shorter, lighter in weight, less expensive, and the right hand tone holes are smaller in diameter and located closer together…Often, a manually-challenged player will find an ergonomically-designed keyless instrument…much more manageable than a larger keyed instrument…
…the great majority of players who experience difficulty in playing tenor and larger recorders are usually convinced that they have small hands and that that is the source of their problems. Most of the time, the difficulty is due not to having small hands, regardless of what they believe or have been told, but rather to incorrect arm, hand, and finger position…
…it should be borne in mind that tenor recorder designs vary a great deal more ergonomically than soprano or alto designs; the hole size and spacing can and do vary greatly from one make and model to another. Players are best advised to seek the help of a dealer with a large and varied inventory of tenor recorders and extensive experience in recorder pedagogy when selecting an instrument to suit their specific needs. There is no one make and model tenor recorder that is universally suitable.